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The intersection graph of graded submodules of a graded module

The intersection graph of graded submodules of a graded module 1IntroductionStudies of graphs associated with algebraic structures developed remarkably in recent years. Usually, the purpose of associating a graph with an algebraic structure is to investigate the algebraic properties using concepts in graph theory. Zero-divisors graph, total graphs, annihilating-ideal graph, and unit graphs are very interesting examples of graphs associated with rings, see [1,2, 3,4]. For studies on graphs associated with graded rings and graded modules, in particular, see [5,6].Among the types of graphs associated with rings are intersection graphs. In 2009, Chakrabarty et al. [7] introduced and studied the intersection graph of ideals of a ring RR, which is an undirected simple graph, denoted by G(R)G\left(R), whose vertices are the nontrivial left ideals of RRand two vertices IIand JJare adjacent if their intersection is nonzero. Inspired by their work, Akbari et al. [8] introduced the intersection graph of submodules of a module. For a ring RRwith unity and a unitary left RR-module MM, the set of all RR-submodules of MMis denoted by S∗(M){S}^{\ast }\left(M). The intersection graph of submodules of MM, denoted by G(M)G\left(M), is an undirected simple graph defined on S∗(M){S}^{\ast }\left(M), where two non-trivial submodules are adjacent if they have a nonzero intersection. Since they were introduced, intersection graphs of ideal and submodules have attracted many researchers to study their graph-theoretic properties and investigate their structures (see [9,10,11, 12,13,14, 15,16,17]). Alraqad et al. [18] introduced and studied the intersection graph of graded ideals of a graded ring.Motivated by all previous works, we introduce the intersection graph of graded submodules of a graded module. Let GGbe a group. A ring RRis said to be GG-graded if there exist additive subgroups {Rσ∣σ∈G}\left\{{R}_{\sigma }| \sigma \in G\right\}such that R=⊕σ∈GRσR={\oplus }_{\sigma \in G}{R}_{\sigma }and RσRτ⊆Rστ{R}_{\sigma }{R}_{\tau }\subseteq {R}_{\sigma \tau }for all σ,τ∈G\sigma ,\tau \in G. A left RR-module MMis said to be GG-graded if there exist additive subgroups Mσ{M}_{\sigma }of MMindexed by the elements σ∈G\sigma \in Gsuch that M=⊕σ∈GMσM={\oplus }_{\sigma \in G}{M}_{\sigma }and RτMσ⊆Mτσ{R}_{\tau }{M}_{\sigma }\subseteq {M}_{\tau \sigma }for all τ,σ∈G\tau ,\sigma \in G. The elements of Mσ{M}_{\sigma }are called homogeneous of degree σ\sigma . If x∈Mx\in M, then xxcan be written uniquely as ∑σ∈Gxσ{\sum }_{\sigma \in G}{x}_{\sigma }, where xσ{x}_{\sigma }is the component of xxin Mσ{M}_{\sigma }. An RR-submodule NNof MMis called GG-graded provided that N=⊕σ∈G(N∩Mσ)N={\oplus }_{\sigma \in G}\left(N\cap {M}_{\sigma }). We denote by hS∗(M)h{S}^{\ast }\left(M)the set of all nontrivial GG-graded RR-submodules of MM.Definition 1.1Let RRbe a GG-graded ring and MMbe a GG-graded left RR-module. The intersection graph of GG-graded submodules of MM, denoted by Γ(G,R,M)\Gamma \left(G,R,M), is defined to be an undirected simple graph whose set of vertices is hS∗(M)h{S}^{\ast }\left(M)and two vertices NNand KKare adjacent if N∩K≠{0}N\cap K\ne \left\{0\right\}.We aim to study the properties of these graphs analogous to the nongraded case. In addition, we investigate connections and relationships among G(Mσ)G\left({M}_{\sigma }), Γ(G,R,M)\Gamma \left(G,R,M), and G(M)G\left(M)under certain types of gradings.The organization of the paper is as follows: Section 2 is devoted to the study of graph-theoretic properties of Γ(G,R,M)\Gamma \left(G,R,M). We discuss their connectivity, diameter, regularity, completeness, domination numbers, and girth. In Section 3, we investigate the relationships between Γ(G,R,M)\Gamma \left(G,R,M)and G(Mσ)G\left({M}_{\sigma })(where Mσ{M}_{\sigma }is considered as a left Re{R}_{e}-module) under some types of gradings such as faithful grading and strong grading. This section also presents some results regarding the relationship between Γ(G,R,M)\Gamma \left(G,R,M)and G(M)G\left(M)when the grading group is a linearly ordered group.For standard terminology and notion in the graph theory, we refer the reader to the textbook [19]. For a simple graph, Γ\Gamma , the set of vertices and set of edges are denoted by V(Γ)V\left(\Gamma )and E(Γ)E\left(\Gamma ), respectively. The cardinality ∣V(Γ)∣| V\left(\Gamma )| is referred to as the order of Γ\Gamma . If x,y∈V(Γ)x,y\in V\left(\Gamma )are adjacent, we denote that as x↔yx\leftrightarrow y. The neighborhood of a vertex xxis N(x)={y∈V(Γ)∣y↔x}{\mathscr{N}}\left(x)=\{y\in V\left(\Gamma )| y\leftrightarrow x\}, and the degree of xxis deg(x)=∣N(x)∣{\rm{\deg }}\left(x)=| {\mathscr{N}}\left(x)| . The graph Γ\Gamma is said to be regular if all of its vertices have the same degree. A graph is called complete (resp. null) if any pair of its vertices are adjacent (resp. not adjacent). A complete (resp. null) graph with nnvertices is denoted by Kn{K}_{n}(resp. Nn{N}_{n}). A graph is said to be connected if any pair of its vertices is connected by a path.Throughout this article, all rings are associated with unity 1≠01\ne 0, and all modules are left modules. When a ring RRis GG-graded, we denote that by (R,G)\left(R,G). The support of (R,G)\left(R,G)is defined as supp(R,G)={σ∈G:Rσ≠0}{\rm{supp}}\left(R,G)=\left\{\sigma \in G:{R}_{\sigma }\ne 0\right\}. If r∈Rr\in R, then rrcan be written uniquely as ∑σ∈Grσ{\sum }_{\sigma \in G}{r}_{\sigma }, where rσ{r}_{\sigma }is the component of rrin Rσ{R}_{\sigma }. It is well known that Re{R}_{e}is a subring of RRwith 1∈Re1\in {R}_{e}. An ideal IIof RRis said to be GG-graded if I=⊕σ∈G(I∩Rσ)I={\oplus }_{\sigma \in G}\left(I\cap {R}_{\sigma }). Let MMbe a GG-graded RR-module. It is known that Mσ{M}_{\sigma }is Re{R}_{e}-submodule of MMfor all σ∈G\sigma \in G. Also, we write h(M)=∪σ∈GMσh\left(M)={\cup }_{\sigma \in G}{M}_{\sigma }and supp(M,G)={σ∈G:Mσ≠0}{\rm{supp}}\left(M,G)=\{\sigma \in G:{M}_{\sigma }\ne 0\}. A GG-graded RR-submodule of MMis said to be GG-graded maximal (resp. simple or minimal) if it is maximal (resp. minimal) among all proper (resp. nonzero) GG-graded RR-submodules of MM. We denote by GMax(M){\rm{GMax}}\left(M)(resp. GMin(M){\rm{GMin}}\left(M)) the set of all nontrivial GG-graded maximal (resp. simple) RR-submodules. A GG-graded RR-module MMis called GG-graded local (resp. GG-graded simple) if ∣GMax(M)∣=1| {\rm{GMax}}\left(M)| =1(resp. GMax(M)={(0)}{\rm{GMax}}\left(M)=\left\{\left(0)\right\}). We say that MMis GG-graded left Noetherian (resp. Artinian) if MMsatisfies the ascending (resp. descending) chain condition for the GG-graded RR-submodules of MM.2Graph theoretic properties of Γ(G,R,M)\Gamma \left(G,R,M)We present the following well-known technical lemma in this section.Lemma 2.1[20, Lemma 2.1] Let RRbe a GG-graded ring and MMbe a GG-graded RR-module. (i)If IIand JJare GG-graded ideals of RR, then I+JI+Jand I⋂JI\hspace{0.33em}\bigcap \hspace{0.33em}Jare GG-graded ideals of RR.(ii)If NNand KKare GG-graded RR-submodules of MM, then N+KN+Kand N⋂KN\hspace{0.33em}\bigcap \hspace{0.33em}Kare GG-graded RR-submodules of MM.(iii)If NNis a GG-graded RR-submodule of MM, r∈h(R)r\in h\left(R), x∈h(M)x\in h\left(M)and IIis a GG-graded ideal of RR, then RxRx, ININ, and rNrNare GG-graded RR-submodules of MM. Moreover, (N:RM)={r∈R:rM⊆N}\left(N{:}_{R}M)=\{r\in R:rM\subseteq N\}is a GG-graded ideal of RR.The following two results from [8] classify disconnected intersection graphs of submodules.Theorem 2.2[8, Theorem 2.1] Let MMbe an RR-module. Then, the graph G(M)G\left(M)is disconnected if and only if MMis a direct sum of two simple RR-modules.Corollary 2.3[8, Corollary 2.3] Let MMbe an RR-module. Then, the graph G(M)G\left(M)is disconnected if and only if it is null graph with at least two vertices.Analogues to the nongraded case, next we characterize disconnected intersection graphs of graded submodules.Theorem 2.4Let RRbe a GG-graded ring and MMbe a GG-graded RR-module such that ∣Γ(G,R,M)∣≥2| \Gamma \left(G,R,M)| \ge 2. Then, the followings are equivalent: (1)Γ(G,R,M)\Gamma \left(G,R,M)is disconnected.(2)Γ(G,R,M)\Gamma \left(G,R,M)is a null graph.(3)Every nontrivial GG-graded RR-submodule of MMis GG-graded maximal as well as GG-graded simple.(4)MMis a direct sum of two GG-graded simple (or maximal) RR-modules.Proof (1)⇒(2)\left(1)\Rightarrow \left(2)Suppose that Γ(G,R,M)\Gamma \left(G,R,M)is disconnected. For a contradiction, assume NNand KKare two adjacent vertices. So NN, KK, and N∩KN\cap Kbelong to the same component of Γ(G,R,M)\Gamma \left(G,R,M). Since Γ(G,R,M)\Gamma \left(G,R,M)is disconnected, there is a vertex LLthat is not connected to any of the vertices NN, KK, and N∩KN\cap K. If (N∩K)+L≠M\left(N\cap K)+L\ne M, then (N∩K)↔((N∩K)+L)↔L\left(N\cap K)\leftrightarrow \left(\left(N\cap K)+L)\leftrightarrow Lis a path connecting N∩KN\cap Kand LL, a contradiction. So (N∩K)+L=M\left(N\cap K)+L=M. Now let a∈Na\in N. Then, a=t+ca=t+cfor some t∈N∩Kt\in N\cap Kand c∈Lc\in L. So a−t=c∈N∩L={0}a-t=c\in N\cap L=\left\{0\right\}; consequently, a=t∈N∩Ka=t\in N\cap K. This implies that N=N∩KN=N\cap K. Similarly, we obtain K=N∩KK=N\cap K. Hence, we have N=KN=K, a contradiction. Therefore, Γ(G,R,M)\Gamma \left(G,R,M)contains no edges, and hence, it is a null graph.(2)⇔(3)\left(2)\iff \left(3)Straightforward.(3)⇒(4)\left(3)\Rightarrow \left(4)Let NNand KKbe GG-graded maximal as well as GG-graded simple RR-submodules of MM. Then, N+K=MN+K=Mand N∩K={0}N\cap K=\left\{0\right\}. Hence, NNand KKare GG-graded simple RR-modules and M=N⊕KM=N\oplus K.(4)⇒(1)\left(4)\Rightarrow \left(1)Suppose M=N⊕KM=N\oplus K, where NNand KKare GG-graded simple RR-modules. Then, NNand KKare GG-graded simple RR-submodules. Also, they are GG-graded maximal because N≅MKN\cong \frac{M}{K}and K≅MNK\cong \frac{M}{N}. Thus, NNand KKare isolated vertices. Therefore, Γ(G,R,M)\Gamma \left(G,R,M)is disconnected.□Roshan-Shekalgourabi and Hassanzadeh-Lelekaami [6] associated a graph GM{G}_{M}with a GG-graded RR-module MM, where V(GM)=hS∗(M)V\left({G}_{M})=h{S}^{\ast }\left(M)and two nontrivial GG-graded RR-submodules NNand KKare adjacent if N+K=MN+K=M. Clearly, the two concepts GM{G}_{M}and Γ(G,R,M)\Gamma \left(G,R,M)are distinct. The next theorem presents an obvious relation between these two graphs.Corollary 2.5Let MMbe a GG-graded RR-module. Then, Γ(G,R,M)\Gamma \left(G,R,M)is disconnected if and only if GM{G}_{M}is a complete graph with at least two vertices.ProofThe result follows by Theorem 2.4 and [6, Theorem 2.2].□The proof of the next corollary is straightforward.Corollary 2.6Let MMbe a GG-graded RR-module. If Γ(G,R,M)\Gamma \left(G,R,M)is connected, then every pair of GG-graded maximal RR-submodules intersect nontrivially.The distance d(x,y)d\left(x,y)between any two vertices x,yx,yin a graph Γ\Gamma is the length of the shortest path between them, and diam(Γ){\rm{diam}}\left(\Gamma )is the supremum of {d(x,y)∣x,y∈V(Γ)}\left\{d\left(x,y)| x,y\in V\left(\Gamma )\right\}.Theorem 2.7Let MMbe a GG-graded RR-module. If Γ(G,R,M)\Gamma \left(G,R,M)is connected, then diam(Γ(G,R,M))≤2{\rm{diam}}\left(\Gamma \left(G,R,M))\le 2.ProofSuppose NNand KKare distinct vertices in Γ(G,R,M)\Gamma \left(G,R,M). If NNand KKare adjacent, then d(N,K)=1d\left(N,K)=1. If NNand KKare nonadjacent, then d(N,K)≥2d\left(N,K)\ge 2. If N⊕K≠MN\oplus K\ne M, then we have the path N↔N⊕K↔KN\leftrightarrow N\oplus K\leftrightarrow K, and hence, d(N,K)=2d\left(N,K)=2. If N⊕K=MN\oplus K=M, then either NNor KKis not GG-graded simple, say NN. Let (0)≠L⊊N\left(0)\ne L\hspace{0.33em}\subsetneq \hspace{0.33em}N. Thus, we have the path N↔L⊕K↔KN\leftrightarrow L\oplus K\leftrightarrow K, and hence, d(N,K)=2d\left(N,K)=2. As a result, d(N,K)≤2d\left(N,K)\le 2.□Theorem 2.8Let MMbe a GG-graded Artinian RR-module such that Γ(G,R,M)\Gamma \left(G,R,M)is not a null graph. Then, the followings are equivalent: (i)Γ(G,R,M)\Gamma \left(G,R,M)is regular.(ii)∣GMin(M)∣=1| {\rm{GMin}}\left(M)| =1.(iii)Γ(G,R,M)\Gamma \left(G,R,M)is complete.Proof (i)⇒(ii)\left(i)\Rightarrow \left(ii)Suppose Γ(G,R,M)\Gamma \left(G,R,M)is regular. Assume that MMcontains two distinct GG-graded simple RR-submodules NNand KK. Clearly, NNand KKare nonadjacent. By Theorem 2.7, there is a GG-graded RR-submodule YYthat is adjacent to both NNand KK. Hence, by minimality of NN, we obtain N⊊YN\hspace{0.33em}\subsetneq \hspace{0.33em}Y. This implies that N(N)⊊N(Y){\mathscr{N}}\left(N)\hspace{0.33em}\subsetneq \hspace{0.33em}{\mathscr{N}}\left(Y); consequently, deg(Y)>deg(N){\rm{\deg }}\left(Y)\gt {\rm{\deg }}\left(N), a contradiction. Hence, MMcontains a unique GG-graded simple RR-submodule.(ii)⇒(iii)\left(ii)\Rightarrow \left(iii)Suppose MMcontains a unique GG-graded simple RR-submodule, say NN. Since MMis GG-graded Artinian, N⊆KN\subseteq Kfor all K∈hS∗(M)K\in h{S}^{\ast }\left(M). Thus, Γ(G,R,M)\Gamma \left(G,R,M)is complete.(iii)⇒(i)\left(iii)\Rightarrow \left(i)Straightforward.□Remark 2.9In a GG-graded RR-module MM, a GG-graded submodule NNof MMis called GG-graded essential if N∩K≠(0)N\cap K\ne \left(0)for all K∈hS∗(M)K\in h{S}^{\ast }\left(M). The graded socle, Gsoc(M){\rm{Gsoc}}\left(M), of MMis the sum of all GG-graded simple RR-submodules of MM. Equivalently Gsoc(M){\rm{Gsoc}}\left(M)equals the intersection of all GG-graded essential RR-submodules of MM, see [21, page 48]. So, if MMis GG-graded Artinian and Γ(G,R,M)\Gamma \left(G,R,M)is complete, then every GG-graded RR-submodule is GG-essential, and thus, by Theorem 2.8, GMin(M)=Gsoc(M){\rm{GMin}}\left(M)={\rm{Gsoc}}\left(M).Recall that the girth of a graph Γ\Gamma , denoted by g(Γ)g\left(\Gamma ), is the length of its shortest cycle. If Γ\Gamma has no cycles, then g(Γ)=∞g\left(\Gamma )=\infty .Theorem 2.10If MMis a GG-graded RR-module, then gr(Γ(G,R,M))∈{3,∞}gr\left(\Gamma \left(G,R,M))\in \left\{3,\infty \right\}.ProofAssume g(Γ(G,R,M))<∞g\left(\Gamma \left(G,R,M))\lt \infty and g(Γ(G,R,M))≥4g\left(\Gamma \left(G,R,M))\ge 4. This implies that every pair of distinct nontrivial GG-graded submodules of MMwith nonzero intersection should be comparable, otherwise Γ(G,R,M)\Gamma \left(G,R,M)will have a cycle of length 3, a contradiction. Since g(Γ(G,R,M))≥4g\left(\Gamma \left(G,R,M))\ge 4, Γ(G,R,M)\Gamma \left(G,R,M)contains a path of length 3, say N↔L↔K↔PN\leftrightarrow L\leftrightarrow K\leftrightarrow P. Since any two submodules in this path are comparable and any chain of nontrivial GG-graded submodules of length 2 induces a cycle of length 3 in Γ(G,R,M)\Gamma \left(G,R,M), the only possible two cases are N⊆LN\subseteq L, K⊆LK\subseteq L, K⊆PK\subseteq Por L⊆NL\subseteq N, L⊆KL\subseteq K, P⊆KP\subseteq K. The first case yields K⊆L∩PK\subseteq L\cap P, and hence, L∩P≠(0)L\cap P\ne \left(0). Thus, L↔K↔P↔LL\leftrightarrow K\leftrightarrow P\leftrightarrow Lis a cycle of length 3 in Γ(G,R,M)\Gamma \left(G,R,M), a contradiction. In the second case, we have (0)≠L⊆N∩K\left(0)\ne L\subseteq N\cap K, and therefore, N↔L↔K↔NN\leftrightarrow L\leftrightarrow K\leftrightarrow Nis a cycle of length 3 in Γ(G,R,M)\Gamma \left(G,R,M), which again yields a contradiction. Therefore, g(Γ(G,R,M))=3g\left(\Gamma \left(G,R,M))=3.□The next theorem gives a characterization of GG-graded RR-modules such that g(Γ(G,R,M))=∞g\left(\Gamma \left(G,R,M))=\infty . Recall that a graph is called star if it has no cycles and has one vertex (the center) that is adjacent to all other vertices.Theorem 2.11Let MMbe a GG-graded Noetherian RR-module such that Γ(G,R,M)\Gamma \left(G,R,M)is not a null graph with ∣Γ(G,R,M)∣≥2| \Gamma \left(G,R,M)| \ge 2. If g(Γ(G,R,M))=∞g\left(\Gamma \left(G,R,M))=\infty , then MMis a GG-graded local module and Γ(G,R,M)\Gamma \left(G,R,M)is a star graph whose center is the unique GG-graded maximal RR-submodule of MM.ProofBy Theorem 2.4, Γ(G,R,M)\Gamma \left(G,R,M)is connected. Suppose that N1{N}_{1}and N2{N}_{2}are two distinct GG-graded maximal RR-submodules of MM. By Theorem 2.7, d(N1,N2)≤2d\left({N}_{1},{N}_{2})\le 2. If N1∩N2≠(0){N}_{1}\cap {N}_{2}\ne \left(0), then N1↔(N1∩N2)↔N2↔N1{N}_{1}\leftrightarrow \left({N}_{1}\cap {N}_{2})\leftrightarrow {N}_{2}\leftrightarrow {N}_{1}is a 3-cycle, a contradiction. So N1∩N2=(0){N}_{1}\cap {N}_{2}=\left(0). Since N1{N}_{1}and N2{N}_{2}are GG-graded maximal RR-submodules, we obtain M=N1⊕N2M={N}_{1}\oplus {N}_{2}. Thus, Γ(G,R,M)\Gamma \left(G,R,M)is null, which contradicts the assumption that Γ(G,R,M)\Gamma \left(G,R,M)is not null. Therefore, MMis GG-graded local module. Let NNbe the GG-graded maximal submodule of MM. It is easy to see that every proper graded submodule of a GG-graded Noetherian module is contained in a GG-graded maximal submodule. So N∩K=K≠(0)N\cap K=K\ne \left(0)for all K∈hS∗(M)K\in h{S}^{\ast }\left(M). However, since Γ(G,R,M)\Gamma \left(G,R,M)has no cycles, we conclude that Γ(G,R,M)\Gamma \left(G,R,M)is a star graph.□A subgraph ϒ\Upsilon of a graph Γ\Gamma is called an induced subgraph if any edge in Γ\Gamma that joins two vertices in ϒ\Upsilon is in ϒ\Upsilon . A complete induced subgraph of a graph Γ\Gamma is called a clique, and the order of the largest clique in Γ\Gamma , denoted by ω(Γ)\omega \left(\Gamma ), is the clique number of Γ\Gamma .Lemma 2.12Let MMbe a GG-graded RR-module. If ω(Γ(G,R,M))<∞\omega \left(\Gamma \left(G,R,M))\lt \infty , then MMis GG-graded Artinian and GG-graded Noetherian.ProofMembers of any ascending or descending chain of GG-graded RR-submodules form a clique in Γ(G,R,M)\Gamma \left(G,R,M), and hence, the chain is finite.□Corollary 2.13Let MMbe a GG-graded RR-module. If Γ(G,R,M)\Gamma \left(G,R,M)is connected, then ∣GMax(M)∣≤ω(Γ(G,R,M))| {\rm{GMax}}\left(M)| \le \omega \left(\Gamma \left(G,R,M)).ProofIf Γ(G,R,M)\Gamma \left(G,R,M)is connected, then by Corollary 2.6, GMax(M){\rm{GMax}}\left(M)is a clique, and thus, ∣GMax(M)∣≤ω(Γ(G,R,M))| {\rm{GMax}}\left(M)| \le \omega \left(\Gamma \left(G,R,M)).□A subset DDof the set of vertices of a graph Γ\Gamma is called a dominating set in Γ\Gamma if every vertex of Γ\Gamma is in DDor adjacent to a vertex in DD. The domination number of Γ\Gamma , denoted by γ(Γ)\gamma \left(\Gamma ), is the minimum cardinality of a dominating set in Γ\Gamma . In the next theorem, we determine the domination number of Γ(G,R,M)\Gamma \left(G,R,M). In this result, we use the notion of graded decomposable modules. A GG-graded RR-module MMis called GG-graded decomposable, if it is a direct sum of two nontrivial GG-graded RR-submodules. If MMis not GG-graded decomposable, then it is called GG-graded indecomposable.Theorem 2.14Let MMbe a GG-graded RR-module that contains a GG-graded maximal submodule. Then, γ(Γ(G,R,M))≤2\gamma \left(\Gamma \left(G,R,M))\le 2. Furthermore, if MMis GG-graded indecomposable, then γ(Γ(G,R,M))=1\gamma \left(\Gamma \left(G,R,M))=1.ProofLet NNbe a GG-graded maximal RR-submodule of MM. If there exists K∈hS∗(M)K\in h{S}^{\ast }\left(M)such that N∩K=(0)N\cap K=\left(0), then N+K=MN+K=M, and hence, M=N⊕KM=N\oplus K. So the set {N,K}\left\{N,K\right\}is a dominating set, and thus, γ(Γ(G,R,M))≤2\gamma \left(\Gamma \left(G,R,M))\le 2. This proves the first part.If MMis GG-graded indecomposable, then N∩K≠(0)N\cap K\ne \left(0)for all K∈hS∗(M)K\in h{S}^{\ast }\left(M). Consequently, {N}\left\{N\right\}is a dominating set, and hence, γ(Γ(G,R,M))=1\gamma \left(\Gamma \left(G,R,M))=1.□3Intersection graph of types of gradingsIn this section, we study some relationships between Γ(G,R,M)\Gamma \left(G,R,M)and G(Mσ)G\left({M}_{\sigma })and between Γ(G,R,M)\Gamma \left(G,R,M)and G(M)G\left(M). It is well known that if MMis a GG-graded RR-module, then Mσ{M}_{\sigma }is an Re{R}_{e}-module for each σ∈G\sigma \in G. So G(Mσ)G\left({M}_{\sigma })here represents the intersection graph of Re{R}_{e}-submodules of Mσ{M}_{\sigma }. We also note that if Nσ{N}_{\sigma }is an Re{R}_{e}-submodule of Mσ{M}_{\sigma }, then RNσR{N}_{\sigma }is a GG-graded RR-submodule of MMand RNσ∩Mσ=NσR{N}_{\sigma }\cap {M}_{\sigma }={N}_{\sigma }.Theorem 3.1Let MMbe a GG-graded RR-module. If for some σ∈G\sigma \in G, G(Mσ)G\left({M}_{\sigma })is connected with at least two vertices, then Γ(G,R,M)\Gamma \left(G,R,M)is connected, and hence, G(M)G\left(M)is connected.ProofSince G(Mσ)G\left({M}_{\sigma })is connected, it must contain an edge. Let Nσ{N}_{\sigma }, Kσ{K}_{\sigma }be two adjacent vertices in G(Mσ)G\left({M}_{\sigma }). Then, RNσR{N}_{\sigma }and RKσR{K}_{\sigma }are vertices in Γ(G,R,M)\Gamma \left(G,R,M). Moreover, RNσ∩Mσ=NσR{N}_{\sigma }\cap {M}_{\sigma }={N}_{\sigma }and RKσ∩Mσ=KσR{K}_{\sigma }\cap {M}_{\sigma }={K}_{\sigma }, and so, RNσ≠RKσR{N}_{\sigma }\ne R{K}_{\sigma }. In addition, we have {0}≠Nσ∩Kσ⊆RNσ∩RKσ\left\{0\right\}\ne {N}_{\sigma }\cap {K}_{\sigma }\subseteq R{N}_{\sigma }\cap R{K}_{\sigma }. Therefore, Γ(G,R,M)\Gamma \left(G,R,M)is not null, and hence, it is connected. The last part follows from Corollary 2.3 because Γ(G,R,M)\Gamma \left(G,R,M)is a subgraph of G(M)G\left(M).□Remark 3.2The converse of Theorem 3.1 needs not to be true in general. Let R=Z6R={{\mathbb{Z}}}_{6}with trivial Z{\mathbb{Z}}-grading; that is, R0=Z6{R}_{0}={{\mathbb{Z}}}_{6}, and Rk=0{R}_{k}=0, for all k≠0k\ne 0, and choose M=Z6[x]M={{\mathbb{Z}}}_{6}\left[x]as Z6{Z}_{6}-module with grading Mk=Z6xk{M}_{k}={{\mathbb{Z}}}_{6}{x}^{k}, k≥0k\ge 0, and Mk=0{M}_{k}=0, k<0k\lt 0. The Z{\mathbb{Z}}-graded Z6{{\mathbb{Z}}}_{6}-submodules Z6{{\mathbb{Z}}}_{6}and Z6+Z6x{{\mathbb{Z}}}_{6}+{{\mathbb{Z}}}_{6}xare adjacent in Γ(Z,Z6,Z6[x])\Gamma \left({\mathbb{Z}},{{\mathbb{Z}}}_{6},{{\mathbb{Z}}}_{6}\left[x]), and by Theorem 2.4, we have Γ(Z,Z6,Z6[x])\Gamma \left({\mathbb{Z}},{{\mathbb{Z}}}_{6},{{\mathbb{Z}}}_{6}\left[x])is connected. On the other hand, for each k≥0k\ge 0, ⟨2xk⟩\langle 2{x}^{k}\rangle and ⟨3xk⟩\langle 3{x}^{k}\rangle are the only Z6{{\mathbb{Z}}}_{6}-submodules of Z6xk{{\mathbb{Z}}}_{6}{x}^{k}, and their intersection is (0)\left(0). So G(Z6xk)G\left({{\mathbb{Z}}}_{6}{x}^{k})is disconnected for all k≥0k\ge 0.A GG-graded RR-module MMis said to be left σ\sigma -faithful for some σ∈G\sigma \in G, if Rστ−1xτ≠{0}{R}_{\sigma {\tau }^{-1}}{x}_{\tau }\ne \left\{0\right\}for every τ∈G\tau \in G, and every nonzero xτ∈Mτ{x}_{\tau }\in {M}_{\tau }. If MMis left σ\sigma -faithful for all σ∈G\sigma \in G, then it is called left faithful.Lemma 3.3[21, Proposition 2.6.3] A GG-graded RR-module MMis σ\sigma -faithful for some σ∈G\sigma \in Gif and only if N∩Mσ≠{0}N\cap {M}_{\sigma }\ne \left\{0\right\}for all N∈hS∗(M)N\in h{S}^{\ast }\left(M).Let MMbe a GG-graded RR-module. Define the simple graph Γσ(G,R,M){\Gamma }_{\sigma }\left(G,R,M)on hS∗(M)h{S}^{\ast }\left(M), where NNand KKare adjacent only if N∩K∩Mσ≠{0}N\cap K\cap {M}_{\sigma }\ne \left\{0\right\}. We will call this graph the σ\sigma -intersection graph of GG-graded RR-modules of MM. It is clear that if N∩K∩Mσ≠{0}N\cap K\cap {M}_{\sigma }\ne \left\{0\right\}, then N∩K≠{0}N\cap K\ne \left\{0\right\}. So Γσ(G,R,M){\Gamma }_{\sigma }\left(G,R,M)is a subgraph of Γ(G,R,M)\Gamma \left(G,R,M).Theorem 3.4Let MMbe a GG-graded RR-module such that Γ(G,R,M)\Gamma \left(G,R,M)is not null graph. Then, MMis σ\sigma -faithful for some σ∈G\sigma \in Gif and only if the map ϕσ:Γσ(G,R,M)⟶Γ(G,R,M){\phi }_{\sigma }:{\Gamma }_{\sigma }\left(G,R,M)\hspace{0.33em}\longrightarrow \hspace{0.33em}\Gamma \left(G,R,M)defined by ϕ(N)=N\phi \left(N)=Nis a graph isomorphism.ProofSuppose MMis σ\sigma -faithful for some σ∈G\sigma \in G. Clearly, ϕ\phi is a set bijection. Let N,K∈hS∗(M)N,K\in h{S}^{\ast }\left(M)such that N∩K≠{0}N\cap K\ne \left\{0\right\}. Since MMis σ\sigma -faithful, by Lemma 3.3, N∩K∩Mσ≠{0}N\cap K\cap {M}_{\sigma }\ne \left\{0\right\}, which implies that NNand KKare adjacent in Γσ(G,R,M){\Gamma }_{\sigma }\left(G,R,M). Therefore, ϕσ{\phi }_{\sigma }is a graph isomorphism. For the converse, suppose that there exists N∈hS∗(M)N\in h{S}^{\ast }\left(M)such that N∩Mσ={0}N\cap {M}_{\sigma }=\left\{0\right\}. Then, NNis an isolated vertex in Γσ(G,R,M){\Gamma }_{\sigma }\left(G,R,M), which implies that NNis an isolated vertex in Γ(G,R,M)\Gamma \left(G,R,M)because ϕσ{\phi }_{\sigma }is an isomorphism. So Γ(G,R,M)\Gamma \left(G,R,M)is null, a contradiction. Then, N∩Mσ≠{0}N\cap {M}_{\sigma }\ne \left\{0\right\}for all N∈hS∗(M)N\in h{S}^{\ast }\left(M). Hence, by Lemma 3.3, MMis σ\sigma -faithful.□Theorem 3.5Let MMbe a σ\sigma -faithful GG-graded RR-module. If RMσ=MR{M}_{\sigma }=M, then the following assertions hold: (i)Γ(G,R,M)\Gamma \left(G,R,M)is connected if and only if G(Mσ)G\left({M}_{\sigma })is connected.(ii)γ(Γ(G,R,M))=γ(G(Mσ))\gamma \left(\Gamma \left(G,R,M))=\gamma \left(G\left({M}_{\sigma })).(iii)ω(Γ(G,R,M))<∞\omega \left(\Gamma \left(G,R,M))\lt \infty if and only if ω(G(Mσ))<∞\omega \left(G\left({M}_{\sigma }))\lt \infty , and for each Nσ∈S∗(Mσ){N}_{\sigma }\in {S}^{\ast }\left({M}_{\sigma }), the set βNσ={N∈hS∗(M)∣N∩Mσ=Nσ}{\beta }_{{N}_{\sigma }}=\{N\in h{S}^{\ast }\left(M)| N\cap {M}_{\sigma }={N}_{\sigma }\}is finite.Proof (i) The “if” part is Theorem 3.1. For the “only if” part, assume Γ(G,R,M)\Gamma \left(G,R,M)is connected and let NσandKσ{N}_{\sigma }\text{and}\hspace{0.25em}{K}_{\sigma }be two distinct vertices in G(Mσ)G\left({M}_{\sigma }). If RNσ∩RKσ≠{0}R{N}_{\sigma }\cap R{K}_{\sigma }\ne \left\{0\right\}, then by Theorem 3.4, RNσ∩RKσ∩Mσ≠{0}R{N}_{\sigma }\cap R{K}_{\sigma }\cap {M}_{\sigma }\ne \left\{0\right\}. So we have Nσ∩Kσ=RNσ∩Mσ∩RKσ∩Mσ=RNσ∩RKσ∩Mσ≠{0}{N}_{\sigma }\cap {K}_{\sigma }=R{N}_{\sigma }\cap {M}_{\sigma }\cap R{K}_{\sigma }\cap {M}_{\sigma }=R{N}_{\sigma }\cap R{K}_{\sigma }\cap {M}_{\sigma }\ne \left\{0\right\}, and hence, Nσ↔Kσ{N}_{\sigma }\leftrightarrow {K}_{\sigma }is a path. Assume RNσ∩RKσ={0}R{N}_{\sigma }\cap R{K}_{\sigma }=\left\{0\right\}. By Theorem 2.7, there is Y∈hS∗(M)Y\in h{S}^{\ast }\left(M)such that RNσ∩Y≠{0}R{N}_{\sigma }\cap Y\ne \left\{0\right\}and RKσ∩Y≠{0}R{K}_{\sigma }\cap Y\ne \left\{0\right\}. Then, RNσ∩Y∩Mσ≠{0}R{N}_{\sigma }\cap Y\cap {M}_{\sigma }\ne \left\{0\right\}and RKσ∩Y∩Mσ≠{0}R{K}_{\sigma }\cap Y\cap {M}_{\sigma }\ne \left\{0\right\}. Since ϕσ{\phi }_{\sigma }is a graph isomorphism, Nσ∩(Y∩Mσ){N}_{\sigma }\cap \left(Y\cap {M}_{\sigma })and Kσ∩(Y∩Mσ){K}_{\sigma }\cap \left(Y\cap {M}_{\sigma })are nontrivial. Moreover, Y∩Mσ≠MσY\cap {M}_{\sigma }\ne {M}_{\sigma }because RMσ=MR{M}_{\sigma }=M. Hence, we obtain a path connecting Nσ{N}_{\sigma }and Kσ{K}_{\sigma }in G(M)G\left(M). Therefore, G(M)G\left(M)is connected.(ii) Let S⊆S∗(Mσ)S\subseteq {S}^{\ast }\left({M}_{\sigma })be a minimal dominating set in G(Mσ)G\left({M}_{\sigma }), and let S={RNσ∣Nσ∈S}{\mathscr{S}}=\left\{R{N}_{\sigma }| {N}_{\sigma }\in S\right\}. Clearly, ∣S∣=∣S∣| {\mathscr{S}}| =| S| . Let K∈hS∗(M)K\in h{S}^{\ast }\left(M)such that K∉SK\notin {\mathscr{S}}. By Lemma 3.3, we have K∩Mσ≠{0}K\cap {M}_{\sigma }\ne \left\{0\right\}, and hence, K∩Mσ∩Nσ≠{0}K\cap {M}_{\sigma }\cap {N}_{\sigma }\ne \left\{0\right\}for some Nσ∈S{N}_{\sigma }\in S. So we have RNσ∈SR{N}_{\sigma }\in {\mathscr{S}}and KKis adjacent to RNσR{N}_{\sigma }in Γ(G,R,M)\Gamma \left(G,R,M). Hence, S{\mathscr{S}}is a dominating set in Γ(G,R,M)\Gamma \left(G,R,M). Therefore, γ(Γ(G,R,M))≤γ(G(Mσ))\gamma \left(\Gamma \left(G,R,M))\le \gamma \left(G\left({M}_{\sigma })). Now assume S{\mathscr{S}}is a minimal dominating set in Γ(G,R,M)\Gamma \left(G,R,M), and let S={N∩Mσ∣N∈S}S=\left\{N\cap {M}_{\sigma }| N\in {\mathscr{S}}\right\}. Let Kσ∈S∗(Mσ){K}_{\sigma }\in {S}^{\ast }\left({M}_{\sigma })such that Kσ∉S{K}_{\sigma }\notin S. If RKσ∈SR{K}_{\sigma }\in {\mathscr{S}}, then RKσ∩Mσ∈SR{K}_{\sigma }\cap {M}_{\sigma }\in {\mathscr{S}}. We have Kσ∩(RKσ∩Mσ)=Kσ≠(0){K}_{\sigma }\cap \left(R{K}_{\sigma }\cap {M}_{\sigma })={K}_{\sigma }\ne \left(0). So Kσ{K}_{\sigma }is adjacent to RKσ∩Mσ∈SR{K}_{\sigma }\cap {M}_{\sigma }\in {\mathscr{S}}. Now assume RKσ∉SR{K}_{\sigma }\notin {\mathscr{S}}. So there exists N∈SN\in {\mathscr{S}}such that RKσ∩N≠(0)R{K}_{\sigma }\cap N\ne \left(0). Hence, by Theorem 3.4, we obtain (0)≠RKσ∩N∩Mσ⊆Kσ∩(N∩Mσ)\left(0)\ne R{K}_{\sigma }\cap N\cap {M}_{\sigma }\subseteq {K}_{\sigma }\cap \left(N\cap {M}_{\sigma }). Thus, Kσ∩(N∩Mσ)≠(0){K}_{\sigma }\cap \left(N\cap {M}_{\sigma })\ne \left(0), and so SSis a dominating set in G(Mσ)G\left({M}_{\sigma }). So γ(G(Mσ))≤∣S∣≤∣S∣=γ(Γ(G,R,M))\gamma \left(G\left({M}_{\sigma }))\le | S| \le | {\mathscr{S}}| =\gamma \left(\Gamma \left(G,R,M)).(iii) Suppose ω(Γ(G,R,M))<∞\omega \left(\Gamma \left(G,R,M))\lt \infty . Let Nσ∈S∗(Mσ){N}_{\sigma }\in {S}^{\ast }\left({M}_{\sigma }). Since all elements of βNσ{\beta }_{{N}_{\sigma }}contain Nσ{N}_{\sigma }, βNσ{\beta }_{{N}_{\sigma }}is a clique in Γ(G,R,M)\Gamma \left(G,R,M). Hence, ∣βNσ∣≤ω(Γ(G,R,M))<∞| {\beta }_{{N}_{\sigma }}| \le \omega \left(\Gamma \left(G,R,M))\lt \infty . Let CCbe a clique in G(Mσ)G\left({M}_{\sigma }). Then, ∪Nσ∈CβNσ{\cup }_{{N}_{\sigma }\in C}{\beta }_{{N}_{\sigma }}is a clique in Γ(G,R,M)\Gamma \left(G,R,M). Thus, ∪Nσ∈CβNσ{\cup }_{{N}_{\sigma }\in C}{\beta }_{{N}_{\sigma }}is finite, which yields CCitself is finite. Therefore, ω(G(Mσ))<∞\omega \left(G\left({M}_{\sigma }))\lt \infty .For the converse, suppose that ω(G(Mσ))<∞\omega \left(G\left({M}_{\sigma }))\lt \infty , and for each Nσ∈S∗(Mσ){N}_{\sigma }\in {S}^{\ast }\left({M}_{\sigma }), the set βNσ{\beta }_{{N}_{\sigma }}is finite. Let DDbe a clique in Γ(G,R,M)\Gamma \left(G,R,M). Let Λ\Lambda be the set of all Nσ∈S∗(Mσ){N}_{\sigma }\in {S}^{\ast }\left({M}_{\sigma })such that D∩βNσD\cap {\beta }_{{N}_{\sigma }}is nonempty. Clearly, the collection {D∩βNσ∣Nσ∈Λ}\left\{D\cap {\beta }_{{N}_{\sigma }}| {N}_{\sigma }\in \Lambda \right\}is a partition of DD. So D=∪Nσ∈Λ(D∩βNσ)D={\cup }_{{N}_{\sigma }\in \Lambda }\left(D\cap {\beta }_{{N}_{\sigma }}). We note that, by the assumption for the converse, D∩βNσD\cap {\beta }_{{N}_{\sigma }}is finite for all Nσ∈Λ{N}_{\sigma }\in \Lambda . Let Nσ,Kσ∈Λ{N}_{\sigma },{K}_{\sigma }\in \Lambda . Then, there are N,K∈DN,K\in Dsuch that Nσ=N∩Mσ{N}_{\sigma }=N\cap {M}_{\sigma }and Kσ=K∩Mσ{K}_{\sigma }=K\cap {M}_{\sigma }. Since DDbe a clique, N∩K≠(0)N\cap K\ne \left(0). In addition, because the grading is σ\sigma -faithful, by Lemma 3.3, we obtain that (0)≠(N∩K)∩Mσ=Nσ∩Kσ\left(0)\ne \left(N\cap K)\cap {M}_{\sigma }={N}_{\sigma }\cap {K}_{\sigma }. So Λ\Lambda is a clique in G(Mσ)G\left({M}_{\sigma }), and hence, it is finite. This implies that D=∪Nσ∈Λ(D∩βNσ)D={\cup }_{{N}_{\sigma }\in \Lambda }\left(D\cap {\beta }_{{N}_{\sigma }})is finite. We just proved that every clique in Γ(G,R,M)\Gamma \left(G,R,M)is finite. Therefore, ω(Γ(G,R,M))<∞\omega \left(\Gamma \left(G,R,M))\lt \infty .□Corollary 3.6Let MMbe a σ\sigma -faithful GG-graded RR-module such that RMσ=MR{M}_{\sigma }=M. Then, Mσ{M}_{\sigma }is a direct sum of two simple Re{R}_{e}-modules if and only if MMis a direct sum of two GG-graded simple RR-modules.ProofThe proof follows directly from Theorems 2.2, 2.4, and Part (i) of Theorem 3.5.□A grading (R,G)\left(R,G)is called strong (resp. first strong) if 1∈RσRσ−11\in {R}_{\sigma }{R}_{{\sigma }^{-1}}for all σ∈G\sigma \in G(resp. σ∈supp(R,G)\sigma \in {\rm{supp}}\left(R,G)) (see [22,23]). In what follows, the symbol ≤\le means “a subgroup of,” while the symbol ≅\cong means “isomorphic to.”Lemma 3.7[23, Fact 2.5] A grading (R,G)\left(R,G)is first strong if and only if H=supp(R,G)≤GH={\rm{supp}}\left(R,G)\le Gand (R,H)\left(R,H)is strong.Lemma 3.8Let (R,G)\left(R,G)be first strong grading and MMbe a GG-graded RR-module. If supp(M,G)⊆supp(R,G){\rm{supp}}\left(M,G)\subseteq {\rm{supp}}\left(R,G), then Γ(G,R,M)≅G(Mσ)\Gamma \left(G,R,M)\cong G\left({M}_{\sigma })for all σ∈supp(M,G)\sigma \in {\rm{supp}}\left(M,G).ProofFix σ∈supp(M,G)\sigma \in {\rm{supp}}\left(M,G). We claim that if NNis GG-graded RR-submodule of MM, then N=R(N∩Mσ)N=R\left(N\cap {M}_{\sigma }). Let 0≠x∈N∩Mτ0\ne x\in N\cap {M}_{\tau }for some τ∈G\tau \in G. Now σ,τ∈supp(M,G)⊆supp(R,G)\sigma ,\tau \in {\rm{supp}}\left(M,G)\subseteq {\rm{supp}}\left(R,G). Thus, since supp(R,G)≤G{\rm{supp}}\left(R,G)\le G, τσ−1∈supp(R,G)\tau {\sigma }^{-1}\in {\rm{supp}}\left(R,G). So Rτσ−1Rστ−1=Re{R}_{\tau {\sigma }^{-1}}{R}_{\sigma {\tau }^{-1}}={R}_{e}. This implies that 1=∑i=1nrisi1={\sum }_{i=1}^{n}{r}_{i}{s}_{i}for some ri∈Rτσ−1{r}_{i}\in {R}_{\tau {\sigma }^{-1}}and si∈Rστ−1{s}_{i}\in {R}_{\sigma {\tau }^{-1}}. Hence, x=∑i=1nrisixx={\sum }_{i=1}^{n}{r}_{i}{s}_{i}x. Since x∈Mτx\in {M}_{\tau }and si∈Rστ−1{s}_{i}\in {R}_{\sigma {\tau }^{-1}}, six∈Rστ−1Mτ⊆Mσ{s}_{i}x\in {R}_{\sigma {\tau }^{-1}}{M}_{\tau }\subseteq {M}_{\sigma }, for all ii. Also six∈N{s}_{i}x\in Nbecause NNis an RR-submodule. So x∈R(N∩Mσ)x\in R\left(N\cap {M}_{\sigma }). Hence, N∩Mτ⊆R(N∩Mσ)N\cap {M}_{\tau }\subseteq R\left(N\cap {M}_{\sigma })for all τ∈supp(M,G)\tau \in {\rm{supp}}\left(M,G). This implies that R(N∩Mσ)⊆N=⊕τ∈G(N∩Mτ)⊆R(N∩Mσ)R\left(N\cap {M}_{\sigma })\hspace{0.25em}\subseteq N={\oplus }_{\tau \in G}\left(N\cap {M}_{\tau })\subseteq R\left(N\cap {M}_{\sigma }). Hence, N=R(N∩Mσ)N=R\left(N\cap {M}_{\sigma }). From the claim, we conclude that M=R(M∩Mσ)=RMσM=R\left(M\cap {M}_{\sigma })=R{M}_{\sigma }and N∩Mσ≠{0}N\cap {M}_{\sigma }\ne \left\{0\right\}for all N∈hS∗(M)N\in h{S}^{\ast }\left(M). Therefore, the correspondence N→N∩MσN\to N\cap {M}_{\sigma }yields an isomorphism between Γ(G,R,M)\Gamma \left(G,R,M)and G(Mσ)G\left({M}_{\sigma }).□Corollary 3.9Let (R,G)\left(R,G)be strong grading and MMbe a GG-graded RR-module. Then, Γ(G,R,M)≅G(Mσ)\Gamma \left(G,R,M)\cong G\left({M}_{\sigma })for all σ∈G\sigma \in G.Example 3.10Let AAbe a ring, and consider the ring R=M3(A)R={M}_{3}\left(A)and the left RR-module M=M3×1(A)M={M}_{3\times 1}\left(A)with Z2{{\mathbb{Z}}}_{2}-gradings given by R0=AA0AA000A,R1=00A00AAA0.M0=AA0,M1=00A.\begin{array}{rcl}{R}_{0}& =& \left[\begin{array}{ccc}A& A& 0\\ A& A& 0\\ 0& 0& A\end{array}\right],\hspace{1em}{R}_{1}=\left[\begin{array}{ccc}0& 0& A\\ 0& 0& A\\ A& A& 0\end{array}\right].\\ {M}_{0}& =& \left[\begin{array}{c}A\\ A\\ 0\end{array}\right],\hspace{3.675em}{M}_{1}=\left[\begin{array}{c}0\\ 0\\ A\end{array}\right].\end{array}Clearly, (R,Z2)\left(R,{{\mathbb{Z}}}_{2})is strong. So by Corollary 3.9, G(Z2,M3(A),M3×1(A))≅G(M1)G\left({{\mathbb{Z}}}_{2},{M}_{3}\left(A),{M}_{3\times 1}\left(A))\cong G\left({M}_{1}). The nontrivial R0{R}_{0}-submodules of M1{M}_{1}are given as follows: 00I∣I∈I∗(A).\left\{\left[\begin{array}{c}0\\ 0\\ I\end{array}\right]| I\in {I}^{\ast }\left(A)\right\}.Hence, G(Z2,M3(A),M3×1(A))≅G(M1)≅G(A)G\left({{\mathbb{Z}}}_{2},{M}_{3}\left(A),{M}_{3\times 1}\left(A))\cong G\left({M}_{1})\cong G\left(A), where AAis considered as left AA-module.For the remainder of this section, we focus on the relationships between the graph-theoretic properties of Γ(G,R,M)\Gamma \left(G,R,M)and G(M)G\left(M)when the grading group is a linearly ordered group. For details on rings and modules graded by linearly ordered group, see [21, Chapter 5].A linearly ordered group is a group GGequipped with a total ordered relation ≤\le such that for all α,β,δ∈G\alpha ,\beta ,\delta \in G, α≤β\alpha \le \beta implies αδ≤βδ\alpha \delta \le \beta \delta and δα≤δβ\delta \alpha \le \delta \beta .Suppose that MMis GG-graded RR-module where GGis a linearly ordered group. Then, any x∈Mx\in Mcan be written uniquely as x=xσ1+xσ2+…+xσnx={x}_{{\sigma }_{1}}+{x}_{{\sigma }_{2}}+\ldots +{x}_{{\sigma }_{n}}, with σ1<σ2<⋯<σn{\sigma }_{1}\lt {\sigma }_{2}\hspace{0.33em}\lt \cdots \lt {\sigma }_{n}. We call xσn{x}_{{\sigma }_{n}}the homogeneous components of xxof highest degree. For each RR-submodule NNof MM, the GG-graded RR-submodule generated by the homogeneous components of the highest degrees of all elements of NNis denoted by N∼{N}^{ \sim }; that is, yyis one of the generators of N∼{N}^{ \sim }if and only if there exists x=xσ1+xσ2+…+xσn∈Nx={x}_{{\sigma }_{1}}+{x}_{{\sigma }_{2}}+\ldots +{x}_{{\sigma }_{n}}\in N, with σ1<σ2<⋯<σn{\sigma }_{1}\lt {\sigma }_{2}\hspace{0.33em}\lt \cdots \lt {\sigma }_{n}and xσn=y{x}_{{\sigma }_{n}}=y. We have the following result from [21, Lemma 5.3.1, Corollary 5.3.3]Lemma 3.11Let MMbe a GG-graded RR-module, where GGis linearly ordered group and NNand KKare submodules of MM. Then, (i)N=N∼N={N}^{ \sim }if and only if NNis GG-graded RR-submodule.(ii)N∼={0}{N}^{ \sim }=\left\{0\right\}if and only if N={0}N=\left\{0\right\}.(iii)If N⊆KN\subseteq K, then N∼⊆K∼{N}^{ \sim }\subseteq {K}^{ \sim }.(iv)If supp(M,G){\rm{supp}}\left(M,G)is well ordered subset of GGand N⊆KN\subseteq K, then N=KN=Kif and only if N∼=K∼{N}^{ \sim }={K}^{ \sim }Theorem 3.12Let MMbe a GG-graded RR-module, where GGis linearly ordered group. If supp(M,G){\rm{supp}}\left(M,G)is well ordered subset of GG, then Γ(G,R,M)\Gamma \left(G,R,M)is connected if and only if G(M)G\left(M)is connected.ProofIf Γ(G,R,M)\Gamma \left(G,R,M)is connected, then G(M)G\left(M)is not null graph, and therefore, it is connected. For the converse, assume that G(M)G\left(M)is connected and let NNand KKbe adjacent vertices of G(M)G\left(M). Hence, N∩K≠{0}N\cap K\ne \left\{0\right\}. Let J=N∩KJ=N\cap K. Since N≠KN\ne K, either J⊊NJ\hspace{0.33em}\subsetneq \hspace{0.33em}Nor J⊊KJ\hspace{0.33em}\subsetneq \hspace{0.33em}K. Without loss of generality, assume J⊊NJ\hspace{0.33em}\subsetneq \hspace{0.33em}N. Then, by parts (ii)–(iv) of Lemma 3.11, we have {0}≠J∼⊊K∼\left\{0\right\}\ne {J}^{ \sim }\hspace{0.33em}\subsetneq \hspace{0.33em}{K}^{ \sim }. So Γ(G,R,M)\Gamma \left(G,R,M)is not null, and hence, it is connected.□Theorem 3.13Let MMbe a GG-graded RR-module, where GGis a linearly ordered group and supp(M,G){\rm{supp}}\left(M,G)is well ordered subset of GG. If MMis local or not Noetherian, then g(Γ(G,R,M))=g(G(M))g\left(\Gamma \left(G,R,M))=g\left(G\left(M)).ProofClearly, if g(G(M))=∞g\left(G\left(M))=\infty , then g(Γ(G,R,M))=∞g\left(\Gamma \left(G,R,M))=\infty . Assume that g(G(M))<∞g\left(G\left(M))\lt \infty , it follows from [11, Theorem 2.5] that g(G(M))=3g\left(G\left(M))=3. If MMis not Noetherian, then there are three nontrivial RR-submodules N1{N}_{1}, N2{N}_{2}, and N3{N}_{3}such that N1⊊N2⊊N3{N}_{1}\hspace{0.33em}\subsetneq \hspace{0.33em}{N}_{2}\hspace{0.33em}\subsetneq \hspace{0.33em}{N}_{3}. Then, by part (iv) of Lemma 3.11, we obtain that N1∼⊊N2∼⊊N3∼{N}_{1}^{ \sim }\hspace{0.33em}\subsetneq \hspace{0.33em}{N}_{2}^{ \sim }\hspace{0.33em}\subsetneq \hspace{0.33em}{N}_{3}^{ \sim }. Hence, N1∼↔N2∼↔N3∼↔N1∼{N}_{1}^{ \sim }\leftrightarrow {N}_{2}^{ \sim }\leftrightarrow {N}_{3}^{ \sim }\leftrightarrow {N}_{1}^{ \sim }is a 3-cycle in Γ(G,R,M)\Gamma \left(G,R,M). Now suppose that MMis Noetherian and local with maximal RR-submodule NN. This implies that K⊆NK\subseteq Nfor all K∈S∗(M)K\in {S}^{\ast }\left(M). Since G(M)G\left(M)is not a star graph, there are two distinct RR-submodules K,L∈S∗(M)⧹{N}K,L\in {S}^{\ast }\left(M)\setminus \left\{N\right\}such that K∩L≠{0}K\cap L\ne \left\{0\right\}. Without the loss of generality, we may assume that K∩L⊊KK\cap L\hspace{0.33em}\subsetneq \hspace{0.33em}K. So we have {0}≠K∩L⊊K⊊N\left\{0\right\}\ne K\cap L\hspace{0.33em}\subsetneq \hspace{0.33em}K\hspace{0.33em}\subsetneq \hspace{0.33em}N. Again by part (iv) of Lemma 3.11, we obtain the 3-cycle (K∩L)∼↔K∼↔N∼↔(K∩L)∼{\left(K\cap L)}^{ \sim }\leftrightarrow {K}^{ \sim }\leftrightarrow {N}^{ \sim }\leftrightarrow {\left(K\cap L)}^{ \sim }in Γ(G,R,M)\Gamma \left(G,R,M). Therefore, g(Γ(G,R,M))=3g\left(\Gamma \left(G,R,M))=3.□The length of an RR-module MMover RR, denoted by ℓ(M)\ell \left(M), is the supremum of the lengths of chains of RR-submodules of MM.Theorem 3.14Let MMbe a GG-graded RR-module, where GGis a linearly ordered group and supp(R,G){\rm{supp}}\left(R,G)is well ordered subset of GG. If g(G(M))≠g(Γ(G,R,M))g\left(G\left(M))\ne g\left(\Gamma \left(G,R,M)), then the following assertions hold(i)g(G(M))=3g\left(G\left(M))=3and g(Γ(G,R,M))=∞g\left(\Gamma \left(G,R,M))=\infty .(ii)MMis GG-graded local but not local.(iii)The length of MMover RRis ℓ(M)=3\ell \left(M)=3.(iv)For every maximal RR-submodule KK, K∼=N{K}^{ \sim }=N, where NNis the unique GG-graded maximal RR-submodule of MM.(v)If rad(M)={0}{\rm{rad}}\left(M)=\left\{0\right\}, then ω(G(M))=∣Max(M)∣\omega \left(G\left(M))=| {\rm{Max}}\left(M)| . Otherwise, ω(G(M))=∣Max(M)∣+1\omega \left(G\left(M))=| {\rm{Max}}\left(M)| +1. (rad(M){\rm{rad}}\left(M)is the intersection of all maximal RR-submodules of MM.)Proof.(i) Follows directly from the fact that Γ(G,R,M)\Gamma \left(G,R,M)is a subgraph of G(M)G\left(M).(ii) Since g(Γ(G,R,M))=∞g\left(\Gamma \left(G,R,M))=\infty , it follows from Theorem 2.11 that MMis GG-graded local, and since g(G(M))≠g(Γ(G,R,M))g\left(G\left(M))\ne g\left(\Gamma \left(G,R,M)), by Theorem 3.13, MMis not local.(iii) Since MMis not local, ℓ(M)≥3\ell \left(M)\ge 3. Assume there are three RR-submodules N1{N}_{1}, N2{N}_{2}, and N3{N}_{3}such that N1⊊N2⊊N3{N}_{1}\hspace{0.33em}\subsetneq \hspace{0.33em}{N}_{2}\hspace{0.33em}\subsetneq \hspace{0.33em}{N}_{3}. Then, by part (iv) of Lemma 3.11, we obtain that N1∼⊊N2∼⊊N3∼{N}_{1}^{ \sim }\hspace{0.33em}\subsetneq \hspace{0.33em}{N}_{2}^{ \sim }\hspace{0.33em}\subsetneq \hspace{0.33em}{N}_{3}^{ \sim }; consequently, g(Γ(G,R,M))=3g\left(\Gamma \left(G,R,M))=3, a contradiction. So ℓ(M)=3\ell \left(M)=3.(iv) Let KKbe a maximal RR-submodule of MM. Since G(M)G\left(M)is connected, KKis not simple. So there is a nontrivial RR-submodule L⊊KL\hspace{0.33em}\subsetneq \hspace{0.33em}K. So L∼⊊K∼{L}^{ \sim }\hspace{0.33em}\subsetneq \hspace{0.33em}{K}^{ \sim }. Also Γ(G,R,M)\Gamma \left(G,R,M)is a star graph, and hence, K∼=N{K}^{ \sim }=N.(v) Since G(M)G\left(M)is connected, Max(M){\rm{Max}}\left(M)is a clique in G(M)G\left(M). Now from part (iii), we obtain that every RR-submodule is either maximal or simple. Moreover, rad(M)∉Max(M){\rm{rad}}\left(M)\notin {\rm{Max}}\left(M)because MMis not local. Therefore, if rad(M)={0}{\rm{rad}}\left(M)=\left\{0\right\}, then Max(M){\rm{Max}}\left(M)is clique of maximum size, otherwise Max(M)∪{rad(M)}{\rm{Max}}\left(M)\cup \left\{{\rm{rad}}\left(M)\right\}is clique of maximum size.□ http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Open Mathematics de Gruyter

The intersection graph of graded submodules of a graded module

Open Mathematics , Volume 20 (1): 10 – Jan 1, 2022

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de Gruyter
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© 2022 Tariq Alraqad, published by De Gruyter
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2391-5455
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10.1515/math-2022-0005
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Abstract

1IntroductionStudies of graphs associated with algebraic structures developed remarkably in recent years. Usually, the purpose of associating a graph with an algebraic structure is to investigate the algebraic properties using concepts in graph theory. Zero-divisors graph, total graphs, annihilating-ideal graph, and unit graphs are very interesting examples of graphs associated with rings, see [1,2, 3,4]. For studies on graphs associated with graded rings and graded modules, in particular, see [5,6].Among the types of graphs associated with rings are intersection graphs. In 2009, Chakrabarty et al. [7] introduced and studied the intersection graph of ideals of a ring RR, which is an undirected simple graph, denoted by G(R)G\left(R), whose vertices are the nontrivial left ideals of RRand two vertices IIand JJare adjacent if their intersection is nonzero. Inspired by their work, Akbari et al. [8] introduced the intersection graph of submodules of a module. For a ring RRwith unity and a unitary left RR-module MM, the set of all RR-submodules of MMis denoted by S∗(M){S}^{\ast }\left(M). The intersection graph of submodules of MM, denoted by G(M)G\left(M), is an undirected simple graph defined on S∗(M){S}^{\ast }\left(M), where two non-trivial submodules are adjacent if they have a nonzero intersection. Since they were introduced, intersection graphs of ideal and submodules have attracted many researchers to study their graph-theoretic properties and investigate their structures (see [9,10,11, 12,13,14, 15,16,17]). Alraqad et al. [18] introduced and studied the intersection graph of graded ideals of a graded ring.Motivated by all previous works, we introduce the intersection graph of graded submodules of a graded module. Let GGbe a group. A ring RRis said to be GG-graded if there exist additive subgroups {Rσ∣σ∈G}\left\{{R}_{\sigma }| \sigma \in G\right\}such that R=⊕σ∈GRσR={\oplus }_{\sigma \in G}{R}_{\sigma }and RσRτ⊆Rστ{R}_{\sigma }{R}_{\tau }\subseteq {R}_{\sigma \tau }for all σ,τ∈G\sigma ,\tau \in G. A left RR-module MMis said to be GG-graded if there exist additive subgroups Mσ{M}_{\sigma }of MMindexed by the elements σ∈G\sigma \in Gsuch that M=⊕σ∈GMσM={\oplus }_{\sigma \in G}{M}_{\sigma }and RτMσ⊆Mτσ{R}_{\tau }{M}_{\sigma }\subseteq {M}_{\tau \sigma }for all τ,σ∈G\tau ,\sigma \in G. The elements of Mσ{M}_{\sigma }are called homogeneous of degree σ\sigma . If x∈Mx\in M, then xxcan be written uniquely as ∑σ∈Gxσ{\sum }_{\sigma \in G}{x}_{\sigma }, where xσ{x}_{\sigma }is the component of xxin Mσ{M}_{\sigma }. An RR-submodule NNof MMis called GG-graded provided that N=⊕σ∈G(N∩Mσ)N={\oplus }_{\sigma \in G}\left(N\cap {M}_{\sigma }). We denote by hS∗(M)h{S}^{\ast }\left(M)the set of all nontrivial GG-graded RR-submodules of MM.Definition 1.1Let RRbe a GG-graded ring and MMbe a GG-graded left RR-module. The intersection graph of GG-graded submodules of MM, denoted by Γ(G,R,M)\Gamma \left(G,R,M), is defined to be an undirected simple graph whose set of vertices is hS∗(M)h{S}^{\ast }\left(M)and two vertices NNand KKare adjacent if N∩K≠{0}N\cap K\ne \left\{0\right\}.We aim to study the properties of these graphs analogous to the nongraded case. In addition, we investigate connections and relationships among G(Mσ)G\left({M}_{\sigma }), Γ(G,R,M)\Gamma \left(G,R,M), and G(M)G\left(M)under certain types of gradings.The organization of the paper is as follows: Section 2 is devoted to the study of graph-theoretic properties of Γ(G,R,M)\Gamma \left(G,R,M). We discuss their connectivity, diameter, regularity, completeness, domination numbers, and girth. In Section 3, we investigate the relationships between Γ(G,R,M)\Gamma \left(G,R,M)and G(Mσ)G\left({M}_{\sigma })(where Mσ{M}_{\sigma }is considered as a left Re{R}_{e}-module) under some types of gradings such as faithful grading and strong grading. This section also presents some results regarding the relationship between Γ(G,R,M)\Gamma \left(G,R,M)and G(M)G\left(M)when the grading group is a linearly ordered group.For standard terminology and notion in the graph theory, we refer the reader to the textbook [19]. For a simple graph, Γ\Gamma , the set of vertices and set of edges are denoted by V(Γ)V\left(\Gamma )and E(Γ)E\left(\Gamma ), respectively. The cardinality ∣V(Γ)∣| V\left(\Gamma )| is referred to as the order of Γ\Gamma . If x,y∈V(Γ)x,y\in V\left(\Gamma )are adjacent, we denote that as x↔yx\leftrightarrow y. The neighborhood of a vertex xxis N(x)={y∈V(Γ)∣y↔x}{\mathscr{N}}\left(x)=\{y\in V\left(\Gamma )| y\leftrightarrow x\}, and the degree of xxis deg(x)=∣N(x)∣{\rm{\deg }}\left(x)=| {\mathscr{N}}\left(x)| . The graph Γ\Gamma is said to be regular if all of its vertices have the same degree. A graph is called complete (resp. null) if any pair of its vertices are adjacent (resp. not adjacent). A complete (resp. null) graph with nnvertices is denoted by Kn{K}_{n}(resp. Nn{N}_{n}). A graph is said to be connected if any pair of its vertices is connected by a path.Throughout this article, all rings are associated with unity 1≠01\ne 0, and all modules are left modules. When a ring RRis GG-graded, we denote that by (R,G)\left(R,G). The support of (R,G)\left(R,G)is defined as supp(R,G)={σ∈G:Rσ≠0}{\rm{supp}}\left(R,G)=\left\{\sigma \in G:{R}_{\sigma }\ne 0\right\}. If r∈Rr\in R, then rrcan be written uniquely as ∑σ∈Grσ{\sum }_{\sigma \in G}{r}_{\sigma }, where rσ{r}_{\sigma }is the component of rrin Rσ{R}_{\sigma }. It is well known that Re{R}_{e}is a subring of RRwith 1∈Re1\in {R}_{e}. An ideal IIof RRis said to be GG-graded if I=⊕σ∈G(I∩Rσ)I={\oplus }_{\sigma \in G}\left(I\cap {R}_{\sigma }). Let MMbe a GG-graded RR-module. It is known that Mσ{M}_{\sigma }is Re{R}_{e}-submodule of MMfor all σ∈G\sigma \in G. Also, we write h(M)=∪σ∈GMσh\left(M)={\cup }_{\sigma \in G}{M}_{\sigma }and supp(M,G)={σ∈G:Mσ≠0}{\rm{supp}}\left(M,G)=\{\sigma \in G:{M}_{\sigma }\ne 0\}. A GG-graded RR-submodule of MMis said to be GG-graded maximal (resp. simple or minimal) if it is maximal (resp. minimal) among all proper (resp. nonzero) GG-graded RR-submodules of MM. We denote by GMax(M){\rm{GMax}}\left(M)(resp. GMin(M){\rm{GMin}}\left(M)) the set of all nontrivial GG-graded maximal (resp. simple) RR-submodules. A GG-graded RR-module MMis called GG-graded local (resp. GG-graded simple) if ∣GMax(M)∣=1| {\rm{GMax}}\left(M)| =1(resp. GMax(M)={(0)}{\rm{GMax}}\left(M)=\left\{\left(0)\right\}). We say that MMis GG-graded left Noetherian (resp. Artinian) if MMsatisfies the ascending (resp. descending) chain condition for the GG-graded RR-submodules of MM.2Graph theoretic properties of Γ(G,R,M)\Gamma \left(G,R,M)We present the following well-known technical lemma in this section.Lemma 2.1[20, Lemma 2.1] Let RRbe a GG-graded ring and MMbe a GG-graded RR-module. (i)If IIand JJare GG-graded ideals of RR, then I+JI+Jand I⋂JI\hspace{0.33em}\bigcap \hspace{0.33em}Jare GG-graded ideals of RR.(ii)If NNand KKare GG-graded RR-submodules of MM, then N+KN+Kand N⋂KN\hspace{0.33em}\bigcap \hspace{0.33em}Kare GG-graded RR-submodules of MM.(iii)If NNis a GG-graded RR-submodule of MM, r∈h(R)r\in h\left(R), x∈h(M)x\in h\left(M)and IIis a GG-graded ideal of RR, then RxRx, ININ, and rNrNare GG-graded RR-submodules of MM. Moreover, (N:RM)={r∈R:rM⊆N}\left(N{:}_{R}M)=\{r\in R:rM\subseteq N\}is a GG-graded ideal of RR.The following two results from [8] classify disconnected intersection graphs of submodules.Theorem 2.2[8, Theorem 2.1] Let MMbe an RR-module. Then, the graph G(M)G\left(M)is disconnected if and only if MMis a direct sum of two simple RR-modules.Corollary 2.3[8, Corollary 2.3] Let MMbe an RR-module. Then, the graph G(M)G\left(M)is disconnected if and only if it is null graph with at least two vertices.Analogues to the nongraded case, next we characterize disconnected intersection graphs of graded submodules.Theorem 2.4Let RRbe a GG-graded ring and MMbe a GG-graded RR-module such that ∣Γ(G,R,M)∣≥2| \Gamma \left(G,R,M)| \ge 2. Then, the followings are equivalent: (1)Γ(G,R,M)\Gamma \left(G,R,M)is disconnected.(2)Γ(G,R,M)\Gamma \left(G,R,M)is a null graph.(3)Every nontrivial GG-graded RR-submodule of MMis GG-graded maximal as well as GG-graded simple.(4)MMis a direct sum of two GG-graded simple (or maximal) RR-modules.Proof (1)⇒(2)\left(1)\Rightarrow \left(2)Suppose that Γ(G,R,M)\Gamma \left(G,R,M)is disconnected. For a contradiction, assume NNand KKare two adjacent vertices. So NN, KK, and N∩KN\cap Kbelong to the same component of Γ(G,R,M)\Gamma \left(G,R,M). Since Γ(G,R,M)\Gamma \left(G,R,M)is disconnected, there is a vertex LLthat is not connected to any of the vertices NN, KK, and N∩KN\cap K. If (N∩K)+L≠M\left(N\cap K)+L\ne M, then (N∩K)↔((N∩K)+L)↔L\left(N\cap K)\leftrightarrow \left(\left(N\cap K)+L)\leftrightarrow Lis a path connecting N∩KN\cap Kand LL, a contradiction. So (N∩K)+L=M\left(N\cap K)+L=M. Now let a∈Na\in N. Then, a=t+ca=t+cfor some t∈N∩Kt\in N\cap Kand c∈Lc\in L. So a−t=c∈N∩L={0}a-t=c\in N\cap L=\left\{0\right\}; consequently, a=t∈N∩Ka=t\in N\cap K. This implies that N=N∩KN=N\cap K. Similarly, we obtain K=N∩KK=N\cap K. Hence, we have N=KN=K, a contradiction. Therefore, Γ(G,R,M)\Gamma \left(G,R,M)contains no edges, and hence, it is a null graph.(2)⇔(3)\left(2)\iff \left(3)Straightforward.(3)⇒(4)\left(3)\Rightarrow \left(4)Let NNand KKbe GG-graded maximal as well as GG-graded simple RR-submodules of MM. Then, N+K=MN+K=Mand N∩K={0}N\cap K=\left\{0\right\}. Hence, NNand KKare GG-graded simple RR-modules and M=N⊕KM=N\oplus K.(4)⇒(1)\left(4)\Rightarrow \left(1)Suppose M=N⊕KM=N\oplus K, where NNand KKare GG-graded simple RR-modules. Then, NNand KKare GG-graded simple RR-submodules. Also, they are GG-graded maximal because N≅MKN\cong \frac{M}{K}and K≅MNK\cong \frac{M}{N}. Thus, NNand KKare isolated vertices. Therefore, Γ(G,R,M)\Gamma \left(G,R,M)is disconnected.□Roshan-Shekalgourabi and Hassanzadeh-Lelekaami [6] associated a graph GM{G}_{M}with a GG-graded RR-module MM, where V(GM)=hS∗(M)V\left({G}_{M})=h{S}^{\ast }\left(M)and two nontrivial GG-graded RR-submodules NNand KKare adjacent if N+K=MN+K=M. Clearly, the two concepts GM{G}_{M}and Γ(G,R,M)\Gamma \left(G,R,M)are distinct. The next theorem presents an obvious relation between these two graphs.Corollary 2.5Let MMbe a GG-graded RR-module. Then, Γ(G,R,M)\Gamma \left(G,R,M)is disconnected if and only if GM{G}_{M}is a complete graph with at least two vertices.ProofThe result follows by Theorem 2.4 and [6, Theorem 2.2].□The proof of the next corollary is straightforward.Corollary 2.6Let MMbe a GG-graded RR-module. If Γ(G,R,M)\Gamma \left(G,R,M)is connected, then every pair of GG-graded maximal RR-submodules intersect nontrivially.The distance d(x,y)d\left(x,y)between any two vertices x,yx,yin a graph Γ\Gamma is the length of the shortest path between them, and diam(Γ){\rm{diam}}\left(\Gamma )is the supremum of {d(x,y)∣x,y∈V(Γ)}\left\{d\left(x,y)| x,y\in V\left(\Gamma )\right\}.Theorem 2.7Let MMbe a GG-graded RR-module. If Γ(G,R,M)\Gamma \left(G,R,M)is connected, then diam(Γ(G,R,M))≤2{\rm{diam}}\left(\Gamma \left(G,R,M))\le 2.ProofSuppose NNand KKare distinct vertices in Γ(G,R,M)\Gamma \left(G,R,M). If NNand KKare adjacent, then d(N,K)=1d\left(N,K)=1. If NNand KKare nonadjacent, then d(N,K)≥2d\left(N,K)\ge 2. If N⊕K≠MN\oplus K\ne M, then we have the path N↔N⊕K↔KN\leftrightarrow N\oplus K\leftrightarrow K, and hence, d(N,K)=2d\left(N,K)=2. If N⊕K=MN\oplus K=M, then either NNor KKis not GG-graded simple, say NN. Let (0)≠L⊊N\left(0)\ne L\hspace{0.33em}\subsetneq \hspace{0.33em}N. Thus, we have the path N↔L⊕K↔KN\leftrightarrow L\oplus K\leftrightarrow K, and hence, d(N,K)=2d\left(N,K)=2. As a result, d(N,K)≤2d\left(N,K)\le 2.□Theorem 2.8Let MMbe a GG-graded Artinian RR-module such that Γ(G,R,M)\Gamma \left(G,R,M)is not a null graph. Then, the followings are equivalent: (i)Γ(G,R,M)\Gamma \left(G,R,M)is regular.(ii)∣GMin(M)∣=1| {\rm{GMin}}\left(M)| =1.(iii)Γ(G,R,M)\Gamma \left(G,R,M)is complete.Proof (i)⇒(ii)\left(i)\Rightarrow \left(ii)Suppose Γ(G,R,M)\Gamma \left(G,R,M)is regular. Assume that MMcontains two distinct GG-graded simple RR-submodules NNand KK. Clearly, NNand KKare nonadjacent. By Theorem 2.7, there is a GG-graded RR-submodule YYthat is adjacent to both NNand KK. Hence, by minimality of NN, we obtain N⊊YN\hspace{0.33em}\subsetneq \hspace{0.33em}Y. This implies that N(N)⊊N(Y){\mathscr{N}}\left(N)\hspace{0.33em}\subsetneq \hspace{0.33em}{\mathscr{N}}\left(Y); consequently, deg(Y)>deg(N){\rm{\deg }}\left(Y)\gt {\rm{\deg }}\left(N), a contradiction. Hence, MMcontains a unique GG-graded simple RR-submodule.(ii)⇒(iii)\left(ii)\Rightarrow \left(iii)Suppose MMcontains a unique GG-graded simple RR-submodule, say NN. Since MMis GG-graded Artinian, N⊆KN\subseteq Kfor all K∈hS∗(M)K\in h{S}^{\ast }\left(M). Thus, Γ(G,R,M)\Gamma \left(G,R,M)is complete.(iii)⇒(i)\left(iii)\Rightarrow \left(i)Straightforward.□Remark 2.9In a GG-graded RR-module MM, a GG-graded submodule NNof MMis called GG-graded essential if N∩K≠(0)N\cap K\ne \left(0)for all K∈hS∗(M)K\in h{S}^{\ast }\left(M). The graded socle, Gsoc(M){\rm{Gsoc}}\left(M), of MMis the sum of all GG-graded simple RR-submodules of MM. Equivalently Gsoc(M){\rm{Gsoc}}\left(M)equals the intersection of all GG-graded essential RR-submodules of MM, see [21, page 48]. So, if MMis GG-graded Artinian and Γ(G,R,M)\Gamma \left(G,R,M)is complete, then every GG-graded RR-submodule is GG-essential, and thus, by Theorem 2.8, GMin(M)=Gsoc(M){\rm{GMin}}\left(M)={\rm{Gsoc}}\left(M).Recall that the girth of a graph Γ\Gamma , denoted by g(Γ)g\left(\Gamma ), is the length of its shortest cycle. If Γ\Gamma has no cycles, then g(Γ)=∞g\left(\Gamma )=\infty .Theorem 2.10If MMis a GG-graded RR-module, then gr(Γ(G,R,M))∈{3,∞}gr\left(\Gamma \left(G,R,M))\in \left\{3,\infty \right\}.ProofAssume g(Γ(G,R,M))<∞g\left(\Gamma \left(G,R,M))\lt \infty and g(Γ(G,R,M))≥4g\left(\Gamma \left(G,R,M))\ge 4. This implies that every pair of distinct nontrivial GG-graded submodules of MMwith nonzero intersection should be comparable, otherwise Γ(G,R,M)\Gamma \left(G,R,M)will have a cycle of length 3, a contradiction. Since g(Γ(G,R,M))≥4g\left(\Gamma \left(G,R,M))\ge 4, Γ(G,R,M)\Gamma \left(G,R,M)contains a path of length 3, say N↔L↔K↔PN\leftrightarrow L\leftrightarrow K\leftrightarrow P. Since any two submodules in this path are comparable and any chain of nontrivial GG-graded submodules of length 2 induces a cycle of length 3 in Γ(G,R,M)\Gamma \left(G,R,M), the only possible two cases are N⊆LN\subseteq L, K⊆LK\subseteq L, K⊆PK\subseteq Por L⊆NL\subseteq N, L⊆KL\subseteq K, P⊆KP\subseteq K. The first case yields K⊆L∩PK\subseteq L\cap P, and hence, L∩P≠(0)L\cap P\ne \left(0). Thus, L↔K↔P↔LL\leftrightarrow K\leftrightarrow P\leftrightarrow Lis a cycle of length 3 in Γ(G,R,M)\Gamma \left(G,R,M), a contradiction. In the second case, we have (0)≠L⊆N∩K\left(0)\ne L\subseteq N\cap K, and therefore, N↔L↔K↔NN\leftrightarrow L\leftrightarrow K\leftrightarrow Nis a cycle of length 3 in Γ(G,R,M)\Gamma \left(G,R,M), which again yields a contradiction. Therefore, g(Γ(G,R,M))=3g\left(\Gamma \left(G,R,M))=3.□The next theorem gives a characterization of GG-graded RR-modules such that g(Γ(G,R,M))=∞g\left(\Gamma \left(G,R,M))=\infty . Recall that a graph is called star if it has no cycles and has one vertex (the center) that is adjacent to all other vertices.Theorem 2.11Let MMbe a GG-graded Noetherian RR-module such that Γ(G,R,M)\Gamma \left(G,R,M)is not a null graph with ∣Γ(G,R,M)∣≥2| \Gamma \left(G,R,M)| \ge 2. If g(Γ(G,R,M))=∞g\left(\Gamma \left(G,R,M))=\infty , then MMis a GG-graded local module and Γ(G,R,M)\Gamma \left(G,R,M)is a star graph whose center is the unique GG-graded maximal RR-submodule of MM.ProofBy Theorem 2.4, Γ(G,R,M)\Gamma \left(G,R,M)is connected. Suppose that N1{N}_{1}and N2{N}_{2}are two distinct GG-graded maximal RR-submodules of MM. By Theorem 2.7, d(N1,N2)≤2d\left({N}_{1},{N}_{2})\le 2. If N1∩N2≠(0){N}_{1}\cap {N}_{2}\ne \left(0), then N1↔(N1∩N2)↔N2↔N1{N}_{1}\leftrightarrow \left({N}_{1}\cap {N}_{2})\leftrightarrow {N}_{2}\leftrightarrow {N}_{1}is a 3-cycle, a contradiction. So N1∩N2=(0){N}_{1}\cap {N}_{2}=\left(0). Since N1{N}_{1}and N2{N}_{2}are GG-graded maximal RR-submodules, we obtain M=N1⊕N2M={N}_{1}\oplus {N}_{2}. Thus, Γ(G,R,M)\Gamma \left(G,R,M)is null, which contradicts the assumption that Γ(G,R,M)\Gamma \left(G,R,M)is not null. Therefore, MMis GG-graded local module. Let NNbe the GG-graded maximal submodule of MM. It is easy to see that every proper graded submodule of a GG-graded Noetherian module is contained in a GG-graded maximal submodule. So N∩K=K≠(0)N\cap K=K\ne \left(0)for all K∈hS∗(M)K\in h{S}^{\ast }\left(M). However, since Γ(G,R,M)\Gamma \left(G,R,M)has no cycles, we conclude that Γ(G,R,M)\Gamma \left(G,R,M)is a star graph.□A subgraph ϒ\Upsilon of a graph Γ\Gamma is called an induced subgraph if any edge in Γ\Gamma that joins two vertices in ϒ\Upsilon is in ϒ\Upsilon . A complete induced subgraph of a graph Γ\Gamma is called a clique, and the order of the largest clique in Γ\Gamma , denoted by ω(Γ)\omega \left(\Gamma ), is the clique number of Γ\Gamma .Lemma 2.12Let MMbe a GG-graded RR-module. If ω(Γ(G,R,M))<∞\omega \left(\Gamma \left(G,R,M))\lt \infty , then MMis GG-graded Artinian and GG-graded Noetherian.ProofMembers of any ascending or descending chain of GG-graded RR-submodules form a clique in Γ(G,R,M)\Gamma \left(G,R,M), and hence, the chain is finite.□Corollary 2.13Let MMbe a GG-graded RR-module. If Γ(G,R,M)\Gamma \left(G,R,M)is connected, then ∣GMax(M)∣≤ω(Γ(G,R,M))| {\rm{GMax}}\left(M)| \le \omega \left(\Gamma \left(G,R,M)).ProofIf Γ(G,R,M)\Gamma \left(G,R,M)is connected, then by Corollary 2.6, GMax(M){\rm{GMax}}\left(M)is a clique, and thus, ∣GMax(M)∣≤ω(Γ(G,R,M))| {\rm{GMax}}\left(M)| \le \omega \left(\Gamma \left(G,R,M)).□A subset DDof the set of vertices of a graph Γ\Gamma is called a dominating set in Γ\Gamma if every vertex of Γ\Gamma is in DDor adjacent to a vertex in DD. The domination number of Γ\Gamma , denoted by γ(Γ)\gamma \left(\Gamma ), is the minimum cardinality of a dominating set in Γ\Gamma . In the next theorem, we determine the domination number of Γ(G,R,M)\Gamma \left(G,R,M). In this result, we use the notion of graded decomposable modules. A GG-graded RR-module MMis called GG-graded decomposable, if it is a direct sum of two nontrivial GG-graded RR-submodules. If MMis not GG-graded decomposable, then it is called GG-graded indecomposable.Theorem 2.14Let MMbe a GG-graded RR-module that contains a GG-graded maximal submodule. Then, γ(Γ(G,R,M))≤2\gamma \left(\Gamma \left(G,R,M))\le 2. Furthermore, if MMis GG-graded indecomposable, then γ(Γ(G,R,M))=1\gamma \left(\Gamma \left(G,R,M))=1.ProofLet NNbe a GG-graded maximal RR-submodule of MM. If there exists K∈hS∗(M)K\in h{S}^{\ast }\left(M)such that N∩K=(0)N\cap K=\left(0), then N+K=MN+K=M, and hence, M=N⊕KM=N\oplus K. So the set {N,K}\left\{N,K\right\}is a dominating set, and thus, γ(Γ(G,R,M))≤2\gamma \left(\Gamma \left(G,R,M))\le 2. This proves the first part.If MMis GG-graded indecomposable, then N∩K≠(0)N\cap K\ne \left(0)for all K∈hS∗(M)K\in h{S}^{\ast }\left(M). Consequently, {N}\left\{N\right\}is a dominating set, and hence, γ(Γ(G,R,M))=1\gamma \left(\Gamma \left(G,R,M))=1.□3Intersection graph of types of gradingsIn this section, we study some relationships between Γ(G,R,M)\Gamma \left(G,R,M)and G(Mσ)G\left({M}_{\sigma })and between Γ(G,R,M)\Gamma \left(G,R,M)and G(M)G\left(M). It is well known that if MMis a GG-graded RR-module, then Mσ{M}_{\sigma }is an Re{R}_{e}-module for each σ∈G\sigma \in G. So G(Mσ)G\left({M}_{\sigma })here represents the intersection graph of Re{R}_{e}-submodules of Mσ{M}_{\sigma }. We also note that if Nσ{N}_{\sigma }is an Re{R}_{e}-submodule of Mσ{M}_{\sigma }, then RNσR{N}_{\sigma }is a GG-graded RR-submodule of MMand RNσ∩Mσ=NσR{N}_{\sigma }\cap {M}_{\sigma }={N}_{\sigma }.Theorem 3.1Let MMbe a GG-graded RR-module. If for some σ∈G\sigma \in G, G(Mσ)G\left({M}_{\sigma })is connected with at least two vertices, then Γ(G,R,M)\Gamma \left(G,R,M)is connected, and hence, G(M)G\left(M)is connected.ProofSince G(Mσ)G\left({M}_{\sigma })is connected, it must contain an edge. Let Nσ{N}_{\sigma }, Kσ{K}_{\sigma }be two adjacent vertices in G(Mσ)G\left({M}_{\sigma }). Then, RNσR{N}_{\sigma }and RKσR{K}_{\sigma }are vertices in Γ(G,R,M)\Gamma \left(G,R,M). Moreover, RNσ∩Mσ=NσR{N}_{\sigma }\cap {M}_{\sigma }={N}_{\sigma }and RKσ∩Mσ=KσR{K}_{\sigma }\cap {M}_{\sigma }={K}_{\sigma }, and so, RNσ≠RKσR{N}_{\sigma }\ne R{K}_{\sigma }. In addition, we have {0}≠Nσ∩Kσ⊆RNσ∩RKσ\left\{0\right\}\ne {N}_{\sigma }\cap {K}_{\sigma }\subseteq R{N}_{\sigma }\cap R{K}_{\sigma }. Therefore, Γ(G,R,M)\Gamma \left(G,R,M)is not null, and hence, it is connected. The last part follows from Corollary 2.3 because Γ(G,R,M)\Gamma \left(G,R,M)is a subgraph of G(M)G\left(M).□Remark 3.2The converse of Theorem 3.1 needs not to be true in general. Let R=Z6R={{\mathbb{Z}}}_{6}with trivial Z{\mathbb{Z}}-grading; that is, R0=Z6{R}_{0}={{\mathbb{Z}}}_{6}, and Rk=0{R}_{k}=0, for all k≠0k\ne 0, and choose M=Z6[x]M={{\mathbb{Z}}}_{6}\left[x]as Z6{Z}_{6}-module with grading Mk=Z6xk{M}_{k}={{\mathbb{Z}}}_{6}{x}^{k}, k≥0k\ge 0, and Mk=0{M}_{k}=0, k<0k\lt 0. The Z{\mathbb{Z}}-graded Z6{{\mathbb{Z}}}_{6}-submodules Z6{{\mathbb{Z}}}_{6}and Z6+Z6x{{\mathbb{Z}}}_{6}+{{\mathbb{Z}}}_{6}xare adjacent in Γ(Z,Z6,Z6[x])\Gamma \left({\mathbb{Z}},{{\mathbb{Z}}}_{6},{{\mathbb{Z}}}_{6}\left[x]), and by Theorem 2.4, we have Γ(Z,Z6,Z6[x])\Gamma \left({\mathbb{Z}},{{\mathbb{Z}}}_{6},{{\mathbb{Z}}}_{6}\left[x])is connected. On the other hand, for each k≥0k\ge 0, ⟨2xk⟩\langle 2{x}^{k}\rangle and ⟨3xk⟩\langle 3{x}^{k}\rangle are the only Z6{{\mathbb{Z}}}_{6}-submodules of Z6xk{{\mathbb{Z}}}_{6}{x}^{k}, and their intersection is (0)\left(0). So G(Z6xk)G\left({{\mathbb{Z}}}_{6}{x}^{k})is disconnected for all k≥0k\ge 0.A GG-graded RR-module MMis said to be left σ\sigma -faithful for some σ∈G\sigma \in G, if Rστ−1xτ≠{0}{R}_{\sigma {\tau }^{-1}}{x}_{\tau }\ne \left\{0\right\}for every τ∈G\tau \in G, and every nonzero xτ∈Mτ{x}_{\tau }\in {M}_{\tau }. If MMis left σ\sigma -faithful for all σ∈G\sigma \in G, then it is called left faithful.Lemma 3.3[21, Proposition 2.6.3] A GG-graded RR-module MMis σ\sigma -faithful for some σ∈G\sigma \in Gif and only if N∩Mσ≠{0}N\cap {M}_{\sigma }\ne \left\{0\right\}for all N∈hS∗(M)N\in h{S}^{\ast }\left(M).Let MMbe a GG-graded RR-module. Define the simple graph Γσ(G,R,M){\Gamma }_{\sigma }\left(G,R,M)on hS∗(M)h{S}^{\ast }\left(M), where NNand KKare adjacent only if N∩K∩Mσ≠{0}N\cap K\cap {M}_{\sigma }\ne \left\{0\right\}. We will call this graph the σ\sigma -intersection graph of GG-graded RR-modules of MM. It is clear that if N∩K∩Mσ≠{0}N\cap K\cap {M}_{\sigma }\ne \left\{0\right\}, then N∩K≠{0}N\cap K\ne \left\{0\right\}. So Γσ(G,R,M){\Gamma }_{\sigma }\left(G,R,M)is a subgraph of Γ(G,R,M)\Gamma \left(G,R,M).Theorem 3.4Let MMbe a GG-graded RR-module such that Γ(G,R,M)\Gamma \left(G,R,M)is not null graph. Then, MMis σ\sigma -faithful for some σ∈G\sigma \in Gif and only if the map ϕσ:Γσ(G,R,M)⟶Γ(G,R,M){\phi }_{\sigma }:{\Gamma }_{\sigma }\left(G,R,M)\hspace{0.33em}\longrightarrow \hspace{0.33em}\Gamma \left(G,R,M)defined by ϕ(N)=N\phi \left(N)=Nis a graph isomorphism.ProofSuppose MMis σ\sigma -faithful for some σ∈G\sigma \in G. Clearly, ϕ\phi is a set bijection. Let N,K∈hS∗(M)N,K\in h{S}^{\ast }\left(M)such that N∩K≠{0}N\cap K\ne \left\{0\right\}. Since MMis σ\sigma -faithful, by Lemma 3.3, N∩K∩Mσ≠{0}N\cap K\cap {M}_{\sigma }\ne \left\{0\right\}, which implies that NNand KKare adjacent in Γσ(G,R,M){\Gamma }_{\sigma }\left(G,R,M). Therefore, ϕσ{\phi }_{\sigma }is a graph isomorphism. For the converse, suppose that there exists N∈hS∗(M)N\in h{S}^{\ast }\left(M)such that N∩Mσ={0}N\cap {M}_{\sigma }=\left\{0\right\}. Then, NNis an isolated vertex in Γσ(G,R,M){\Gamma }_{\sigma }\left(G,R,M), which implies that NNis an isolated vertex in Γ(G,R,M)\Gamma \left(G,R,M)because ϕσ{\phi }_{\sigma }is an isomorphism. So Γ(G,R,M)\Gamma \left(G,R,M)is null, a contradiction. Then, N∩Mσ≠{0}N\cap {M}_{\sigma }\ne \left\{0\right\}for all N∈hS∗(M)N\in h{S}^{\ast }\left(M). Hence, by Lemma 3.3, MMis σ\sigma -faithful.□Theorem 3.5Let MMbe a σ\sigma -faithful GG-graded RR-module. If RMσ=MR{M}_{\sigma }=M, then the following assertions hold: (i)Γ(G,R,M)\Gamma \left(G,R,M)is connected if and only if G(Mσ)G\left({M}_{\sigma })is connected.(ii)γ(Γ(G,R,M))=γ(G(Mσ))\gamma \left(\Gamma \left(G,R,M))=\gamma \left(G\left({M}_{\sigma })).(iii)ω(Γ(G,R,M))<∞\omega \left(\Gamma \left(G,R,M))\lt \infty if and only if ω(G(Mσ))<∞\omega \left(G\left({M}_{\sigma }))\lt \infty , and for each Nσ∈S∗(Mσ){N}_{\sigma }\in {S}^{\ast }\left({M}_{\sigma }), the set βNσ={N∈hS∗(M)∣N∩Mσ=Nσ}{\beta }_{{N}_{\sigma }}=\{N\in h{S}^{\ast }\left(M)| N\cap {M}_{\sigma }={N}_{\sigma }\}is finite.Proof (i) The “if” part is Theorem 3.1. For the “only if” part, assume Γ(G,R,M)\Gamma \left(G,R,M)is connected and let NσandKσ{N}_{\sigma }\text{and}\hspace{0.25em}{K}_{\sigma }be two distinct vertices in G(Mσ)G\left({M}_{\sigma }). If RNσ∩RKσ≠{0}R{N}_{\sigma }\cap R{K}_{\sigma }\ne \left\{0\right\}, then by Theorem 3.4, RNσ∩RKσ∩Mσ≠{0}R{N}_{\sigma }\cap R{K}_{\sigma }\cap {M}_{\sigma }\ne \left\{0\right\}. So we have Nσ∩Kσ=RNσ∩Mσ∩RKσ∩Mσ=RNσ∩RKσ∩Mσ≠{0}{N}_{\sigma }\cap {K}_{\sigma }=R{N}_{\sigma }\cap {M}_{\sigma }\cap R{K}_{\sigma }\cap {M}_{\sigma }=R{N}_{\sigma }\cap R{K}_{\sigma }\cap {M}_{\sigma }\ne \left\{0\right\}, and hence, Nσ↔Kσ{N}_{\sigma }\leftrightarrow {K}_{\sigma }is a path. Assume RNσ∩RKσ={0}R{N}_{\sigma }\cap R{K}_{\sigma }=\left\{0\right\}. By Theorem 2.7, there is Y∈hS∗(M)Y\in h{S}^{\ast }\left(M)such that RNσ∩Y≠{0}R{N}_{\sigma }\cap Y\ne \left\{0\right\}and RKσ∩Y≠{0}R{K}_{\sigma }\cap Y\ne \left\{0\right\}. Then, RNσ∩Y∩Mσ≠{0}R{N}_{\sigma }\cap Y\cap {M}_{\sigma }\ne \left\{0\right\}and RKσ∩Y∩Mσ≠{0}R{K}_{\sigma }\cap Y\cap {M}_{\sigma }\ne \left\{0\right\}. Since ϕσ{\phi }_{\sigma }is a graph isomorphism, Nσ∩(Y∩Mσ){N}_{\sigma }\cap \left(Y\cap {M}_{\sigma })and Kσ∩(Y∩Mσ){K}_{\sigma }\cap \left(Y\cap {M}_{\sigma })are nontrivial. Moreover, Y∩Mσ≠MσY\cap {M}_{\sigma }\ne {M}_{\sigma }because RMσ=MR{M}_{\sigma }=M. Hence, we obtain a path connecting Nσ{N}_{\sigma }and Kσ{K}_{\sigma }in G(M)G\left(M). Therefore, G(M)G\left(M)is connected.(ii) Let S⊆S∗(Mσ)S\subseteq {S}^{\ast }\left({M}_{\sigma })be a minimal dominating set in G(Mσ)G\left({M}_{\sigma }), and let S={RNσ∣Nσ∈S}{\mathscr{S}}=\left\{R{N}_{\sigma }| {N}_{\sigma }\in S\right\}. Clearly, ∣S∣=∣S∣| {\mathscr{S}}| =| S| . Let K∈hS∗(M)K\in h{S}^{\ast }\left(M)such that K∉SK\notin {\mathscr{S}}. By Lemma 3.3, we have K∩Mσ≠{0}K\cap {M}_{\sigma }\ne \left\{0\right\}, and hence, K∩Mσ∩Nσ≠{0}K\cap {M}_{\sigma }\cap {N}_{\sigma }\ne \left\{0\right\}for some Nσ∈S{N}_{\sigma }\in S. So we have RNσ∈SR{N}_{\sigma }\in {\mathscr{S}}and KKis adjacent to RNσR{N}_{\sigma }in Γ(G,R,M)\Gamma \left(G,R,M). Hence, S{\mathscr{S}}is a dominating set in Γ(G,R,M)\Gamma \left(G,R,M). Therefore, γ(Γ(G,R,M))≤γ(G(Mσ))\gamma \left(\Gamma \left(G,R,M))\le \gamma \left(G\left({M}_{\sigma })). Now assume S{\mathscr{S}}is a minimal dominating set in Γ(G,R,M)\Gamma \left(G,R,M), and let S={N∩Mσ∣N∈S}S=\left\{N\cap {M}_{\sigma }| N\in {\mathscr{S}}\right\}. Let Kσ∈S∗(Mσ){K}_{\sigma }\in {S}^{\ast }\left({M}_{\sigma })such that Kσ∉S{K}_{\sigma }\notin S. If RKσ∈SR{K}_{\sigma }\in {\mathscr{S}}, then RKσ∩Mσ∈SR{K}_{\sigma }\cap {M}_{\sigma }\in {\mathscr{S}}. We have Kσ∩(RKσ∩Mσ)=Kσ≠(0){K}_{\sigma }\cap \left(R{K}_{\sigma }\cap {M}_{\sigma })={K}_{\sigma }\ne \left(0). So Kσ{K}_{\sigma }is adjacent to RKσ∩Mσ∈SR{K}_{\sigma }\cap {M}_{\sigma }\in {\mathscr{S}}. Now assume RKσ∉SR{K}_{\sigma }\notin {\mathscr{S}}. So there exists N∈SN\in {\mathscr{S}}such that RKσ∩N≠(0)R{K}_{\sigma }\cap N\ne \left(0). Hence, by Theorem 3.4, we obtain (0)≠RKσ∩N∩Mσ⊆Kσ∩(N∩Mσ)\left(0)\ne R{K}_{\sigma }\cap N\cap {M}_{\sigma }\subseteq {K}_{\sigma }\cap \left(N\cap {M}_{\sigma }). Thus, Kσ∩(N∩Mσ)≠(0){K}_{\sigma }\cap \left(N\cap {M}_{\sigma })\ne \left(0), and so SSis a dominating set in G(Mσ)G\left({M}_{\sigma }). So γ(G(Mσ))≤∣S∣≤∣S∣=γ(Γ(G,R,M))\gamma \left(G\left({M}_{\sigma }))\le | S| \le | {\mathscr{S}}| =\gamma \left(\Gamma \left(G,R,M)).(iii) Suppose ω(Γ(G,R,M))<∞\omega \left(\Gamma \left(G,R,M))\lt \infty . Let Nσ∈S∗(Mσ){N}_{\sigma }\in {S}^{\ast }\left({M}_{\sigma }). Since all elements of βNσ{\beta }_{{N}_{\sigma }}contain Nσ{N}_{\sigma }, βNσ{\beta }_{{N}_{\sigma }}is a clique in Γ(G,R,M)\Gamma \left(G,R,M). Hence, ∣βNσ∣≤ω(Γ(G,R,M))<∞| {\beta }_{{N}_{\sigma }}| \le \omega \left(\Gamma \left(G,R,M))\lt \infty . Let CCbe a clique in G(Mσ)G\left({M}_{\sigma }). Then, ∪Nσ∈CβNσ{\cup }_{{N}_{\sigma }\in C}{\beta }_{{N}_{\sigma }}is a clique in Γ(G,R,M)\Gamma \left(G,R,M). Thus, ∪Nσ∈CβNσ{\cup }_{{N}_{\sigma }\in C}{\beta }_{{N}_{\sigma }}is finite, which yields CCitself is finite. Therefore, ω(G(Mσ))<∞\omega \left(G\left({M}_{\sigma }))\lt \infty .For the converse, suppose that ω(G(Mσ))<∞\omega \left(G\left({M}_{\sigma }))\lt \infty , and for each Nσ∈S∗(Mσ){N}_{\sigma }\in {S}^{\ast }\left({M}_{\sigma }), the set βNσ{\beta }_{{N}_{\sigma }}is finite. Let DDbe a clique in Γ(G,R,M)\Gamma \left(G,R,M). Let Λ\Lambda be the set of all Nσ∈S∗(Mσ){N}_{\sigma }\in {S}^{\ast }\left({M}_{\sigma })such that D∩βNσD\cap {\beta }_{{N}_{\sigma }}is nonempty. Clearly, the collection {D∩βNσ∣Nσ∈Λ}\left\{D\cap {\beta }_{{N}_{\sigma }}| {N}_{\sigma }\in \Lambda \right\}is a partition of DD. So D=∪Nσ∈Λ(D∩βNσ)D={\cup }_{{N}_{\sigma }\in \Lambda }\left(D\cap {\beta }_{{N}_{\sigma }}). We note that, by the assumption for the converse, D∩βNσD\cap {\beta }_{{N}_{\sigma }}is finite for all Nσ∈Λ{N}_{\sigma }\in \Lambda . Let Nσ,Kσ∈Λ{N}_{\sigma },{K}_{\sigma }\in \Lambda . Then, there are N,K∈DN,K\in Dsuch that Nσ=N∩Mσ{N}_{\sigma }=N\cap {M}_{\sigma }and Kσ=K∩Mσ{K}_{\sigma }=K\cap {M}_{\sigma }. Since DDbe a clique, N∩K≠(0)N\cap K\ne \left(0). In addition, because the grading is σ\sigma -faithful, by Lemma 3.3, we obtain that (0)≠(N∩K)∩Mσ=Nσ∩Kσ\left(0)\ne \left(N\cap K)\cap {M}_{\sigma }={N}_{\sigma }\cap {K}_{\sigma }. So Λ\Lambda is a clique in G(Mσ)G\left({M}_{\sigma }), and hence, it is finite. This implies that D=∪Nσ∈Λ(D∩βNσ)D={\cup }_{{N}_{\sigma }\in \Lambda }\left(D\cap {\beta }_{{N}_{\sigma }})is finite. We just proved that every clique in Γ(G,R,M)\Gamma \left(G,R,M)is finite. Therefore, ω(Γ(G,R,M))<∞\omega \left(\Gamma \left(G,R,M))\lt \infty .□Corollary 3.6Let MMbe a σ\sigma -faithful GG-graded RR-module such that RMσ=MR{M}_{\sigma }=M. Then, Mσ{M}_{\sigma }is a direct sum of two simple Re{R}_{e}-modules if and only if MMis a direct sum of two GG-graded simple RR-modules.ProofThe proof follows directly from Theorems 2.2, 2.4, and Part (i) of Theorem 3.5.□A grading (R,G)\left(R,G)is called strong (resp. first strong) if 1∈RσRσ−11\in {R}_{\sigma }{R}_{{\sigma }^{-1}}for all σ∈G\sigma \in G(resp. σ∈supp(R,G)\sigma \in {\rm{supp}}\left(R,G)) (see [22,23]). In what follows, the symbol ≤\le means “a subgroup of,” while the symbol ≅\cong means “isomorphic to.”Lemma 3.7[23, Fact 2.5] A grading (R,G)\left(R,G)is first strong if and only if H=supp(R,G)≤GH={\rm{supp}}\left(R,G)\le Gand (R,H)\left(R,H)is strong.Lemma 3.8Let (R,G)\left(R,G)be first strong grading and MMbe a GG-graded RR-module. If supp(M,G)⊆supp(R,G){\rm{supp}}\left(M,G)\subseteq {\rm{supp}}\left(R,G), then Γ(G,R,M)≅G(Mσ)\Gamma \left(G,R,M)\cong G\left({M}_{\sigma })for all σ∈supp(M,G)\sigma \in {\rm{supp}}\left(M,G).ProofFix σ∈supp(M,G)\sigma \in {\rm{supp}}\left(M,G). We claim that if NNis GG-graded RR-submodule of MM, then N=R(N∩Mσ)N=R\left(N\cap {M}_{\sigma }). Let 0≠x∈N∩Mτ0\ne x\in N\cap {M}_{\tau }for some τ∈G\tau \in G. Now σ,τ∈supp(M,G)⊆supp(R,G)\sigma ,\tau \in {\rm{supp}}\left(M,G)\subseteq {\rm{supp}}\left(R,G). Thus, since supp(R,G)≤G{\rm{supp}}\left(R,G)\le G, τσ−1∈supp(R,G)\tau {\sigma }^{-1}\in {\rm{supp}}\left(R,G). So Rτσ−1Rστ−1=Re{R}_{\tau {\sigma }^{-1}}{R}_{\sigma {\tau }^{-1}}={R}_{e}. This implies that 1=∑i=1nrisi1={\sum }_{i=1}^{n}{r}_{i}{s}_{i}for some ri∈Rτσ−1{r}_{i}\in {R}_{\tau {\sigma }^{-1}}and si∈Rστ−1{s}_{i}\in {R}_{\sigma {\tau }^{-1}}. Hence, x=∑i=1nrisixx={\sum }_{i=1}^{n}{r}_{i}{s}_{i}x. Since x∈Mτx\in {M}_{\tau }and si∈Rστ−1{s}_{i}\in {R}_{\sigma {\tau }^{-1}}, six∈Rστ−1Mτ⊆Mσ{s}_{i}x\in {R}_{\sigma {\tau }^{-1}}{M}_{\tau }\subseteq {M}_{\sigma }, for all ii. Also six∈N{s}_{i}x\in Nbecause NNis an RR-submodule. So x∈R(N∩Mσ)x\in R\left(N\cap {M}_{\sigma }). Hence, N∩Mτ⊆R(N∩Mσ)N\cap {M}_{\tau }\subseteq R\left(N\cap {M}_{\sigma })for all τ∈supp(M,G)\tau \in {\rm{supp}}\left(M,G). This implies that R(N∩Mσ)⊆N=⊕τ∈G(N∩Mτ)⊆R(N∩Mσ)R\left(N\cap {M}_{\sigma })\hspace{0.25em}\subseteq N={\oplus }_{\tau \in G}\left(N\cap {M}_{\tau })\subseteq R\left(N\cap {M}_{\sigma }). Hence, N=R(N∩Mσ)N=R\left(N\cap {M}_{\sigma }). From the claim, we conclude that M=R(M∩Mσ)=RMσM=R\left(M\cap {M}_{\sigma })=R{M}_{\sigma }and N∩Mσ≠{0}N\cap {M}_{\sigma }\ne \left\{0\right\}for all N∈hS∗(M)N\in h{S}^{\ast }\left(M). Therefore, the correspondence N→N∩MσN\to N\cap {M}_{\sigma }yields an isomorphism between Γ(G,R,M)\Gamma \left(G,R,M)and G(Mσ)G\left({M}_{\sigma }).□Corollary 3.9Let (R,G)\left(R,G)be strong grading and MMbe a GG-graded RR-module. Then, Γ(G,R,M)≅G(Mσ)\Gamma \left(G,R,M)\cong G\left({M}_{\sigma })for all σ∈G\sigma \in G.Example 3.10Let AAbe a ring, and consider the ring R=M3(A)R={M}_{3}\left(A)and the left RR-module M=M3×1(A)M={M}_{3\times 1}\left(A)with Z2{{\mathbb{Z}}}_{2}-gradings given by R0=AA0AA000A,R1=00A00AAA0.M0=AA0,M1=00A.\begin{array}{rcl}{R}_{0}& =& \left[\begin{array}{ccc}A& A& 0\\ A& A& 0\\ 0& 0& A\end{array}\right],\hspace{1em}{R}_{1}=\left[\begin{array}{ccc}0& 0& A\\ 0& 0& A\\ A& A& 0\end{array}\right].\\ {M}_{0}& =& \left[\begin{array}{c}A\\ A\\ 0\end{array}\right],\hspace{3.675em}{M}_{1}=\left[\begin{array}{c}0\\ 0\\ A\end{array}\right].\end{array}Clearly, (R,Z2)\left(R,{{\mathbb{Z}}}_{2})is strong. So by Corollary 3.9, G(Z2,M3(A),M3×1(A))≅G(M1)G\left({{\mathbb{Z}}}_{2},{M}_{3}\left(A),{M}_{3\times 1}\left(A))\cong G\left({M}_{1}). The nontrivial R0{R}_{0}-submodules of M1{M}_{1}are given as follows: 00I∣I∈I∗(A).\left\{\left[\begin{array}{c}0\\ 0\\ I\end{array}\right]| I\in {I}^{\ast }\left(A)\right\}.Hence, G(Z2,M3(A),M3×1(A))≅G(M1)≅G(A)G\left({{\mathbb{Z}}}_{2},{M}_{3}\left(A),{M}_{3\times 1}\left(A))\cong G\left({M}_{1})\cong G\left(A), where AAis considered as left AA-module.For the remainder of this section, we focus on the relationships between the graph-theoretic properties of Γ(G,R,M)\Gamma \left(G,R,M)and G(M)G\left(M)when the grading group is a linearly ordered group. For details on rings and modules graded by linearly ordered group, see [21, Chapter 5].A linearly ordered group is a group GGequipped with a total ordered relation ≤\le such that for all α,β,δ∈G\alpha ,\beta ,\delta \in G, α≤β\alpha \le \beta implies αδ≤βδ\alpha \delta \le \beta \delta and δα≤δβ\delta \alpha \le \delta \beta .Suppose that MMis GG-graded RR-module where GGis a linearly ordered group. Then, any x∈Mx\in Mcan be written uniquely as x=xσ1+xσ2+…+xσnx={x}_{{\sigma }_{1}}+{x}_{{\sigma }_{2}}+\ldots +{x}_{{\sigma }_{n}}, with σ1<σ2<⋯<σn{\sigma }_{1}\lt {\sigma }_{2}\hspace{0.33em}\lt \cdots \lt {\sigma }_{n}. We call xσn{x}_{{\sigma }_{n}}the homogeneous components of xxof highest degree. For each RR-submodule NNof MM, the GG-graded RR-submodule generated by the homogeneous components of the highest degrees of all elements of NNis denoted by N∼{N}^{ \sim }; that is, yyis one of the generators of N∼{N}^{ \sim }if and only if there exists x=xσ1+xσ2+…+xσn∈Nx={x}_{{\sigma }_{1}}+{x}_{{\sigma }_{2}}+\ldots +{x}_{{\sigma }_{n}}\in N, with σ1<σ2<⋯<σn{\sigma }_{1}\lt {\sigma }_{2}\hspace{0.33em}\lt \cdots \lt {\sigma }_{n}and xσn=y{x}_{{\sigma }_{n}}=y. We have the following result from [21, Lemma 5.3.1, Corollary 5.3.3]Lemma 3.11Let MMbe a GG-graded RR-module, where GGis linearly ordered group and NNand KKare submodules of MM. Then, (i)N=N∼N={N}^{ \sim }if and only if NNis GG-graded RR-submodule.(ii)N∼={0}{N}^{ \sim }=\left\{0\right\}if and only if N={0}N=\left\{0\right\}.(iii)If N⊆KN\subseteq K, then N∼⊆K∼{N}^{ \sim }\subseteq {K}^{ \sim }.(iv)If supp(M,G){\rm{supp}}\left(M,G)is well ordered subset of GGand N⊆KN\subseteq K, then N=KN=Kif and only if N∼=K∼{N}^{ \sim }={K}^{ \sim }Theorem 3.12Let MMbe a GG-graded RR-module, where GGis linearly ordered group. If supp(M,G){\rm{supp}}\left(M,G)is well ordered subset of GG, then Γ(G,R,M)\Gamma \left(G,R,M)is connected if and only if G(M)G\left(M)is connected.ProofIf Γ(G,R,M)\Gamma \left(G,R,M)is connected, then G(M)G\left(M)is not null graph, and therefore, it is connected. For the converse, assume that G(M)G\left(M)is connected and let NNand KKbe adjacent vertices of G(M)G\left(M). Hence, N∩K≠{0}N\cap K\ne \left\{0\right\}. Let J=N∩KJ=N\cap K. Since N≠KN\ne K, either J⊊NJ\hspace{0.33em}\subsetneq \hspace{0.33em}Nor J⊊KJ\hspace{0.33em}\subsetneq \hspace{0.33em}K. Without loss of generality, assume J⊊NJ\hspace{0.33em}\subsetneq \hspace{0.33em}N. Then, by parts (ii)–(iv) of Lemma 3.11, we have {0}≠J∼⊊K∼\left\{0\right\}\ne {J}^{ \sim }\hspace{0.33em}\subsetneq \hspace{0.33em}{K}^{ \sim }. So Γ(G,R,M)\Gamma \left(G,R,M)is not null, and hence, it is connected.□Theorem 3.13Let MMbe a GG-graded RR-module, where GGis a linearly ordered group and supp(M,G){\rm{supp}}\left(M,G)is well ordered subset of GG. If MMis local or not Noetherian, then g(Γ(G,R,M))=g(G(M))g\left(\Gamma \left(G,R,M))=g\left(G\left(M)).ProofClearly, if g(G(M))=∞g\left(G\left(M))=\infty , then g(Γ(G,R,M))=∞g\left(\Gamma \left(G,R,M))=\infty . Assume that g(G(M))<∞g\left(G\left(M))\lt \infty , it follows from [11, Theorem 2.5] that g(G(M))=3g\left(G\left(M))=3. If MMis not Noetherian, then there are three nontrivial RR-submodules N1{N}_{1}, N2{N}_{2}, and N3{N}_{3}such that N1⊊N2⊊N3{N}_{1}\hspace{0.33em}\subsetneq \hspace{0.33em}{N}_{2}\hspace{0.33em}\subsetneq \hspace{0.33em}{N}_{3}. Then, by part (iv) of Lemma 3.11, we obtain that N1∼⊊N2∼⊊N3∼{N}_{1}^{ \sim }\hspace{0.33em}\subsetneq \hspace{0.33em}{N}_{2}^{ \sim }\hspace{0.33em}\subsetneq \hspace{0.33em}{N}_{3}^{ \sim }. Hence, N1∼↔N2∼↔N3∼↔N1∼{N}_{1}^{ \sim }\leftrightarrow {N}_{2}^{ \sim }\leftrightarrow {N}_{3}^{ \sim }\leftrightarrow {N}_{1}^{ \sim }is a 3-cycle in Γ(G,R,M)\Gamma \left(G,R,M). Now suppose that MMis Noetherian and local with maximal RR-submodule NN. This implies that K⊆NK\subseteq Nfor all K∈S∗(M)K\in {S}^{\ast }\left(M). Since G(M)G\left(M)is not a star graph, there are two distinct RR-submodules K,L∈S∗(M)⧹{N}K,L\in {S}^{\ast }\left(M)\setminus \left\{N\right\}such that K∩L≠{0}K\cap L\ne \left\{0\right\}. Without the loss of generality, we may assume that K∩L⊊KK\cap L\hspace{0.33em}\subsetneq \hspace{0.33em}K. So we have {0}≠K∩L⊊K⊊N\left\{0\right\}\ne K\cap L\hspace{0.33em}\subsetneq \hspace{0.33em}K\hspace{0.33em}\subsetneq \hspace{0.33em}N. Again by part (iv) of Lemma 3.11, we obtain the 3-cycle (K∩L)∼↔K∼↔N∼↔(K∩L)∼{\left(K\cap L)}^{ \sim }\leftrightarrow {K}^{ \sim }\leftrightarrow {N}^{ \sim }\leftrightarrow {\left(K\cap L)}^{ \sim }in Γ(G,R,M)\Gamma \left(G,R,M). Therefore, g(Γ(G,R,M))=3g\left(\Gamma \left(G,R,M))=3.□The length of an RR-module MMover RR, denoted by ℓ(M)\ell \left(M), is the supremum of the lengths of chains of RR-submodules of MM.Theorem 3.14Let MMbe a GG-graded RR-module, where GGis a linearly ordered group and supp(R,G){\rm{supp}}\left(R,G)is well ordered subset of GG. If g(G(M))≠g(Γ(G,R,M))g\left(G\left(M))\ne g\left(\Gamma \left(G,R,M)), then the following assertions hold(i)g(G(M))=3g\left(G\left(M))=3and g(Γ(G,R,M))=∞g\left(\Gamma \left(G,R,M))=\infty .(ii)MMis GG-graded local but not local.(iii)The length of MMover RRis ℓ(M)=3\ell \left(M)=3.(iv)For every maximal RR-submodule KK, K∼=N{K}^{ \sim }=N, where NNis the unique GG-graded maximal RR-submodule of MM.(v)If rad(M)={0}{\rm{rad}}\left(M)=\left\{0\right\}, then ω(G(M))=∣Max(M)∣\omega \left(G\left(M))=| {\rm{Max}}\left(M)| . Otherwise, ω(G(M))=∣Max(M)∣+1\omega \left(G\left(M))=| {\rm{Max}}\left(M)| +1. (rad(M){\rm{rad}}\left(M)is the intersection of all maximal RR-submodules of MM.)Proof.(i) Follows directly from the fact that Γ(G,R,M)\Gamma \left(G,R,M)is a subgraph of G(M)G\left(M).(ii) Since g(Γ(G,R,M))=∞g\left(\Gamma \left(G,R,M))=\infty , it follows from Theorem 2.11 that MMis GG-graded local, and since g(G(M))≠g(Γ(G,R,M))g\left(G\left(M))\ne g\left(\Gamma \left(G,R,M)), by Theorem 3.13, MMis not local.(iii) Since MMis not local, ℓ(M)≥3\ell \left(M)\ge 3. Assume there are three RR-submodules N1{N}_{1}, N2{N}_{2}, and N3{N}_{3}such that N1⊊N2⊊N3{N}_{1}\hspace{0.33em}\subsetneq \hspace{0.33em}{N}_{2}\hspace{0.33em}\subsetneq \hspace{0.33em}{N}_{3}. Then, by part (iv) of Lemma 3.11, we obtain that N1∼⊊N2∼⊊N3∼{N}_{1}^{ \sim }\hspace{0.33em}\subsetneq \hspace{0.33em}{N}_{2}^{ \sim }\hspace{0.33em}\subsetneq \hspace{0.33em}{N}_{3}^{ \sim }; consequently, g(Γ(G,R,M))=3g\left(\Gamma \left(G,R,M))=3, a contradiction. So ℓ(M)=3\ell \left(M)=3.(iv) Let KKbe a maximal RR-submodule of MM. Since G(M)G\left(M)is connected, KKis not simple. So there is a nontrivial RR-submodule L⊊KL\hspace{0.33em}\subsetneq \hspace{0.33em}K. So L∼⊊K∼{L}^{ \sim }\hspace{0.33em}\subsetneq \hspace{0.33em}{K}^{ \sim }. Also Γ(G,R,M)\Gamma \left(G,R,M)is a star graph, and hence, K∼=N{K}^{ \sim }=N.(v) Since G(M)G\left(M)is connected, Max(M){\rm{Max}}\left(M)is a clique in G(M)G\left(M). Now from part (iii), we obtain that every RR-submodule is either maximal or simple. Moreover, rad(M)∉Max(M){\rm{rad}}\left(M)\notin {\rm{Max}}\left(M)because MMis not local. Therefore, if rad(M)={0}{\rm{rad}}\left(M)=\left\{0\right\}, then Max(M){\rm{Max}}\left(M)is clique of maximum size, otherwise Max(M)∪{rad(M)}{\rm{Max}}\left(M)\cup \left\{{\rm{rad}}\left(M)\right\}is clique of maximum size.□

Journal

Open Mathematicsde Gruyter

Published: Jan 1, 2022

Keywords: graded ring; graded module; intersection graph; submodule; homogeneous; 13A02; 05C25; 16W50

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