Ideal Structure of Leavitt Path Algebras with Coefficients in a Unital Commutative RingLarki, Hossein
doi: 10.1080/00927872.2014.946133pmid: N/A
For a (countable) graph E and a unital commutative ring R, we analyze the ideal structure of the Leavitt path algebra L R (E) introduced by Mark Tomforde. We first modify the definition of basic ideals and then develop the ideal characterization of Mark Tomforde. We also give necessary and sufficient conditions for the primeness and the primitivity of L R (E), and we then determine prime graded basic ideals and left (or right) primitive graded ideals of L R (E). In particular, when E satisfies Condition (K) and R is a field, they imply that the set of prime ideals and the set of primitive ideals of L R (E) coincide.
On Abelian Groups Having All Proper Fully Invariant Subgroups IsomorphicChekhlov, A. R.; Danchev, P. V.
doi: 10.1080/00927872.2015.1008011pmid: N/A
We introduce two classes of abelian groups which have either only trivial fully invariant subgroups or all their nontrivial (respectively nonzero) fully invariant subgroups are isomorphic, called IFI-groups and strongly IFI-groups, such that every strongly IFI-group is an IFI-group, respectively. Moreover, these classes coincide when the groups are torsion-free, but are different when the groups are torsion as well as, surprisingly, mixed groups cannot be IFI-groups. We also study their important properties as our results somewhat contrast with those from [13] and [14].
Automorphism Groups of Schur RingsKerby, Brent
doi: 10.1080/00927872.2014.952739pmid: N/A
In 1993, Muzychuk [23] showed that the rational Schur rings over a cyclic group Z n are in one-to-one correspondence with sublattices of the divisor lattice of n, or equivalently, with sublattices of the lattice of subgroups of Z n . This can easily be extended to show that for any finite group G, sublattices of the lattice of characteristic subgroups of G give rise to rational Schur rings over G in a natural way. Our main result is that any finite group may be represented as the (algebraic) automorphism group of such a rational Schur ring over an abelian p-group, for any odd prime p. In contrast, over a cyclic group the automorphism group of any Schur ring is abelian. We also prove a converse to the well-known result of Muzychuk [24] that two Schur rings over a cyclic group are isomorphic if and only if they coincide; namely, we show that over a group which is not cyclic, there always exist distinct isomorphic Schur rings.
A New Construction for Cohen–Macaulay GraphsMousivand, Amir; Seyed Fakhari, Seyed
Amin; Yassemi, Siamak
doi: 10.1080/00927872.2014.955576pmid: N/A
Let G be a finite simple graph on a vertex set V(G) = {x 11,…, x n1}. Also let m 1,…, m n ≥ 2 be integers and G 1,…, G n be connected simple graphs on the vertex sets V(G i ) = {x i1,…, x im i }. In this article, we provide necessary and sufficient conditions on G 1,…, G n for which the graph obtained by attaching the G i to G is unmixed or vertex decomposable. Then we characterize Cohen–Macaulay and sequentially Cohen–Macaulay graphs obtained by attaching the cycle graphs or connected chordal graphs to arbitrary graphs.
On NI Skew Polynomial RingsNasr-Isfahani, A. R.
doi: 10.1080/00927872.2014.957385pmid: N/A
Let R be a ring with an endomorphism α and an α-derivation δ. In this article, we first compute the Jacobson radical of NI ℤ-graded rings and show that J(S) = Niℓ(S) if and only if is a ℤ-graded NI ring and J(S) ∩ S 0 is nil. As a corollary we show that, J(R[x; α]) = Niℓ(R[x; α]) if and only if R[x; α] is NI and J(R[x; α]) ∩ R ⊆ Niℓ(R). If R[x, x −1; α] is NI we prove that, J(R[x, x −1; α]) = Niℓ(R[x, x −1; α]) = Niℓ*(R[x, x −1; α]) = Niℓ(R)[x, x −1; α]. We also provide necessary and sufficient conditions for a skew polynomial ring R[x; α, δ] and skew Laurent polynomial ring R[x, x −1; α] to be NI.
The Nakayama Property of a Module and Related ConceptsGottlieb, Christian
doi: 10.1080/00927872.2014.958849pmid: N/A
Three related properties of a module are investigated in this article, namely the Nakayama property, the Maximal property, and the S-property. A module M has the Nakayama property if 𝔞M = M for an ideal 𝔞 implies that sM = 0 for some s ∈ 𝔞 + 1. A module M has the Maximal property if there is in M a maximal proper submodule, and finally, M is said to have the S-property if S −1 M = 0 for a multiplicatively closed set S implies that sM = 0 for some s ∈ S.
Singular and Totally Singular Unitary SpacesDolphin, Andrew
doi: 10.1080/00927872.2014.967353pmid: N/A
In this article, we present a decomposition theorem for unitary spaces over a central simple division algebra with involution in characteristic 2. This is a generalization of a decomposition result for quadratic forms in characteristic 2 from [4] and extends a generalization of the Witt decomposition theorem for nonsingular spaces in this setting to cover spaces that may be singular.
Character Sums for Cayley GraphsAbdollahi, Alireza; Zallaghi, Maysam
doi: 10.1080/00927872.2014.967398pmid: N/A
Following [1], by a Cayley digraph we mean a graph Cay(G, S) whose vertex set is a group G, and there exists a directed edge from a vertex g to another vertex h if g −1 h ∈ S, where S is a generating subset of G. The graph Cay(G, S) is called a Cayley graph if S = S −1 and 1 ∉ S. In Problem 3.3 of the above cited article, the following question is proposed. Let G be a finite group, let Γ = Cay(G, S) be a Cayley digraph, ν a positive integer, and where χ1, …, χ h are all irreducible characters of G. Is the set M ν = {μ i | χ i (1) = ν} an invariant of Γ? (Thus, does Cay(G, S) ≅ Cay(G, S′) imply ?) It is easy to see that the set M ν is an invariant for the Cayley digraphs if the underlying group G is abelian. Here we negatively answer the above problem. We show that for every n ≥ 4 there is a Cayley graph Γ n on the symmetric group S n so that the above set M ν is not an invariant of Γ n . We also find some other groups with the latter property.