On (k, n)-absorbing hyperideals in Krasner (m, n)-hyperrings

On (k, n)-absorbing hyperideals in Krasner (m, n)-hyperrings Abstract Krasner Hyperrings are an important class of algebraic hyperstructures which generalize rings further to allow multiple output values for the addition operation. Krasner (m, n)-hyperrings are a generalization of canonical n-ary hypergroups, a generalization of (m, n)-rings, a generalizations of hyperrings in the sense of Krasner and a subclass of the (m, n)-hyperrings. Badawi introduced a generalization of prime ideals called 2-absorbing ideals and this idea is further generalized in a paper by Anderson and Badawi to a concept called n-absorbing ideals. In this paper, we generalize this notion in Krasner (m, n)-hyperrings introducing and studying the notion of (k, n)-absorbing hyperideal. Several properties of them are provided. Also we introduce the notion of (k, n)-absorbing primary hyperideals of H generalizing the analogs results obtained in rings. Several properties of them are provided. 1. Introduction and preliminaries Algebraic structures play a prominent role in mathematics with wide ranging applications in many disciplines such as theoretical physics, computer sciences, control engineering, information sciences and coding theory. Hyperstructures, as a natural extension of classical algebraic structures, in particular hypergroups, were introduced in 1934 by the French mathematician, Marty, at the 8th Congress of Scandinavian Mathematicians [26]. Since then, a lot of papers and several books have been written on this topic. Nowadays, hyperstructures have a lot of applications to several domains of mathematics and computer science (see [6, 34]) and they are studied in many countries of the world. This theory has been subsequently developed by Corsini [7, 8, 9], Mittas [29, 30], Stratigopoulos [32] and by various authors. Basic definitions and propositions about the hyperstructures are found in [7, 9, 13, 16, 34]. Krasner [22] has studied the notion of hyperfields, hyperrings and then some researchers, namely, Davvaz [14, 15, 19], Vougiouklis [33, 34] and others followed him. In a classical algebraic structure, the composition of two elements is an element, while in an algebraic hyperstructure, the composition of two elements is a set. Several books have been written on hyperstructure theory, see [7, 9, 16, 34]. A recent book on hyperstructures [9] points out on their applications in rough set theory, cryptography, codes, automata, probability, geometry, lattices, binary relations, graphs and hypergraphs. Another book [16] is devoted especially to the study of hyperring theory. Several kinds of hyperrings are introduced and analyzed. The volume ends with an outline of applications in chemistry and physics, analyzing several special kinds of hyperstructures: e-hyperstructures and transposition hypergroups. The theory of suitable modified hyperstructures can serve as a mathematical background in the field of quantum communication systems. n-Ary generalizations of algebraic structures are the most natural way for further development and deeper understanding of their fundamental properties. In [18], Davvaz and Vougiouklis introduced the concept of n-ary hypergroups as a generalization of hypergroups in the sense of Marty. Also, we can consider n-ary hypergroups as a nice generalization of n-ary groups. Leoreanu-Fotea and Davvaz in [25] introduced and studied the notion of a partial n-ary hypergroupoid, associated with a binary relation. Some important results, concerning Rosenberg partial hypergroupoids, induced by relations, are generalized to the case of n-ary hypergroupoids. Davvaz et al. in [17] considered a class of algebraic hypersystems which represent a generalization of semigroups, hypersemigroups and n-ary semigroups. Rings are sets that possess algebraic operations of addition and multiplication of elements in the set, encompassing examples such as the integers and rational numbers. Hyperrings generalize rings further to allow multiple output values for the addition operation. In mathematics, introducing more abstract definitions often serves to clarify and consolidate phenomena that arise in a variety of settings. Hyperrings arise naturally in several settings in algebra, including quadratic form theory, number theory, orderings and ordered algebraic structures, tropical geometry, and multiplicative subgroups of fields. Hyperrings are essentially rings with approximately modified axioms. Hyperrings (R,+,·) are of different types introduced by different researchers. The hyperrings have appeared as a new class of algebraic hyperstructures more general than that of hyperfields, introduced by Krasner [22] in the theory of valued fields. Krasner introduced a type of hyperring (R,+,·) where + is a hyperoperation and · is an ordinary binary operation. Such a hyperring is called a Krasner hyperring. In 2007, Davvaz and Leoreanu-Fotea [16] published a book titled Hyperring Theory and Applications. Several authors generalized the study of ordinary rings to the case of where the ring operations are respectively m-ary and n-ary. (m,n)-rings were studied by Crombez [10], Crombez and Timm [11], Dudek [20] and Lee [23]. Davvaz in [12] and also Mirvakili and Davvaz in [27] defined (m,n)-hyperrings and obtained several results in this respect. Moreover, they [28] introduced Krasner (m,n)-hyperrings as a subclass of (m,n)-hyperrings and as a generalization of Krasner hyperrings. The concept of prime ideal, which arises in the theory of rings as a generalization of the concept of prime number in the ring of integers, plays a highly important role in that theory, as might be expected from the central position occupied by the primes in arithmetic. Badawi [4] introduced a generalization of prime ideals called 2-absorbing ideals and this idea is further generalized by Anderson and Badawi [1, 3] to a concept called n-absorbing ideal. In this paper, we generalize this notion in Krasner (m,n)-hyperrings introducing and studying the notion of (k,n)-absorbing hyperideal. Several properties of them are provided. Also we introduce the notion of (k,n)-absorbing primary hyperideal of H generalizing the analogs results obtained in rings (cf. [5]). Several properties of them are provided. 2. Some basic definitions Recall first the basic terms and definitions from the hyperrings theory. The notion of (m,n)-ary hyperring was introduced by Davvaz [12]. Let H be a nonempty set and f:H×H→*(H) be a mapping, where *(H) denotes the set of all nonempty subsets of H. Then, f is called a binary (algebraic) hyperoperation on H. In general, a mapping f:H×H×⋯×H→*(H), where H appears m times, is called an m-ary (algebraic) hyperoperation, and m is called the arity of this hyperoperation. An algebraic system (H,f), where f is an m-ary hyperoperation defined on H, is called an m-ary hypergroupoid or an m-ary hypersystem. Let f be an m-ary hyperoperation on H and A1,A2,…,Am be nonempty subsets of H. We define   f(A1,A2,…,Am)=⋃{f(x1,x2,…,xm)∣xi∈Ai,i=1,2,…,m}. We shall use the following abbreviated notation: the sequence xi,xi+1,…,xj will be denoted by xij. For j<i, xij is the empty symbol. In this convention,   f(x1,…,xi,yi+1,…,yj,zj+1,…,zm)will be written as f(x1i,yi+1j,zj+1m). In the case when yi+1=⋯=yj=y, the last expression will be written in the form f(x1i,y(j−i),zj+1m). Similarly, for subsets A1,A2,…,Am of H we define   f(A1m)=f(A1,A2,…,Am)=⋃{f(x1m)∣xi∈Ai,i=1,…,m}. An m-ary hyperoperation f is called (i,j)- associative if   f(x1i−1,f(xim+i−1),xm+i2m−1)=f(x1j−1,f(xjm+j−1),xm+j2m−1)holds for fixed 1≤i<j≤m and all x1,x2,…,x2m−1∈H. Note that (i,k)-associativity follows from (i,j)- and (j,k)-associativity. If the above condition is satisfied for all i,j∈{1,2,…,m}, then we say that f is associative. The m-ary hyperoperation f is called commutative iff for all x1,…,xm∈H and for all σ∈Sm,f(x1,x2,…,xm)=f(xσ(1),xσ(2),…,xσ(m)). An m-ary hypergroupoid with the associative m-ary hyperoperation is called an m-ary semihypergroup. An m-ary hypergroupoid (H,f) in which the equation b∈f(a1i−1,xi,ai+1m) has a solution xi∈H for every a1i−1,ai+1m,b∈H and 1≤i≤m is called an m-ary quasihypergroup and when (H,f) is an m-ary semihypergroup, (H,f) is called an m-ary hypergroup. Let B be a nonempty subset of H. Then, B is called an m-ary subhypergroup of (H,f), if f(x1m)⊆B for x1m∈B, and the equation b∈f(b1i−1,xi,bi+1m) has a solution xi∈B for every b1i−1,bi+1m,b∈B and 1≤i≤m. An element e∈H is called a scalar neutral element if x=f(e(i−1),x,e(m−i)), for every 1≤i≤m and for every x∈H. An m-ary polygroup [21] is an m-ary hypergroup (P,f) such that the following axioms hold for all 1≤i,j≤m and x,x1m∈P: There exists a unique element 0∈P such that xi=f(0(i−1),x,0(m−i)). There exists a unitary operation—on P such that x∈f(x1m) implies that xi∈f(−xi−1,…,−x1,x,−xn,…,−xi+1). It is clear that every m-ary group with a scalar neutral element is an m-ary polygroup. A canonical m-ary hypergroup [24] is a commutative m-ary polygroup. An element 0 of an m-ary semihypergroup (H,g) is called a zero element if for every x2n∈H, we have g(0,x2n)=g(x2,0,x3n)=⋯=g(x2n,0)=0. If 0 and 0′ are two zero elements, then 0=g(0′,0(m−1))=0′ and so the zero element is unique. Davvaz in [12] and also Mirvakili and Davvaz in [27] defined (m,n)-hyperrings and obtained several results in this respect. Moreover, they introduced Krasner (m,n)-hyperrings as a subclass of (m,n)-hyperrings and as a generalization of Krasner hyperrings, as follows. Definition 2.1. A Krasner (m,n)-hyperring is an algebraic hyperstructure (R,f,g) which satisfies the following axioms: (R,f) is a cannonical m-ary hypergroup. (R,g) is an n-ary semigroup. The n-ary operation g is distributive with respect to the m-ary hyperoperation f, that is, for every a1i−1,ai+1n,x1m∈R,1≤i≤n,   g(a1i−1,f(x1m),ai+1n)=f(g(a1i−1,x1,ai+1n),…,g(a1i−1,xm,ai+1n)). 0 is a zero element (absorbing element) of the n-ary operation g, that is, for every x2n−1∈R, we have g(0,x2n)=g(x2,0,x3n)=⋯=g(x2n,0)=0. It is clear that every Krasner hyperring is a Krasner (2,2)-hyperring. Also, every Krasner (m,0)-hyperring is a cannonical m-ary hyperring and every Krasner (0,n)-hyperring is an n-ary semigroup. Several examples of Krasner (m,n)-hyperrings can be found in [2, 28, 31]. Let S be a nonempty subset of a Krasner (m,n)-hyperring (R,f,g). If (S,f,g) is a Krasner (m,n)-hyperring, then S is called a subhyperring of R. Let I be a nonempty subset of R and 1≤i≤n; we call I an i-hyperideal of R if I is a subhypergroup of the cannonical m-ary hypergroup (R,f), that is, (I,f) is a cannonical n-ary hypergroup. For every x1n∈R,g(x1i−1,I,xi+1n)⊆I.Also, if for every 1≤i≤n, I is an i-hyperideal, then I is called a hyperideal of R. Every hyperideal of R is a subhyperring of R. A hyperideal P of a Krasner (m,n)-hyperring (R,f,g), such that P≠R, is called an n-ary prime hyperideal if for hyperideals U1,…,Un of R, g(U1n)⊆P implies that U1⊆P or U2⊆P or … or Un⊆P [2]. Equivalently, P is an n-ary prime hyperideal of R if for all a1n∈R, g(a1n)∈P implies a1∈P or … or an∈P. A Krasner (m,n)-hyperring (R,f,g) is commutative if (R,g) is a commutative n-ary semigroup. Also, we say that (R,f,g) is with scalar identity if there exists an element 1g such that x=g(x,1g(n−1)) for all x∈R. A Krasner (m,n)-hyperring (R,f,g) is called an n-ary hyperdomain, if R is a commutative Krasner (m,n)-hyperring with scalar identity and g(a1n)=0 implies that a1=0 or a2=0 or … or an=0 for all a1n∈R. Let I be a hyperideal in a commutative Krasner (m,n)-hyperring (R,f,g) with scalar identity. The radical of I, denoted by I(m,n), is the hyperideal ⋂P, where the intersection is taken over all n-ary prime hyperideals P which contain I. If the set of all n-ary hyperideals containing I is empty, then I(m,n) is defined to be R [2]. By the [2, Theorem 4.23], if I is a hyperideal in a commutative Krasner (m,n)-hyperring (R,f,g) with scalar identity 1g, then I(m,n)={r∈R∣g(r(t),1g(n−t))∈I} for t≤n and I(m,n)={r∈R∣g(l)(r(t))∈I} for t>n,t=l(n−1)+1, for some t∈N. Throughout the paper, we will assume that R is a commutative Krasner (m,n)-hyperring with identity element. 3. On (k,n)-absorbing hyperideals Definition 3.1. Let R be a commutative Krasner (m,n)-hyperring and k a positive integer. A proper hyperideal I of R is an (k,n)-absorbing hyperideal of R if whenever g(x1,…,xkn−k+1)∈I for x1,…,xkn−k+1∈H, then there are (k−1)n−k+2 of the xi’s whose g-product is in I. It is clear that for k=1 we take the n-ary prime hyperideal of Krasner (m,n)-hyperring [2]. For k=1 and n=2, we take the classic prime hyperideal of a Krasner hyperring. Example 3.2. Consider Z12={0,1,2,3,4,5,6,7,8,9,10,11} and the multiplicative group of units Z12*={1,5,7,11} of Z12. The Krasner construction Z12/Z12* is a hyperring. The hyperideals I={0Z12*,3Z12*,6Z12*} and J={0Z12*,2Z12*,4Z12*,6Z12*} are (2,2)-absorbing. Example 3.3. Let H={0,1,a} and B={0,a}. Then, (H,+,·) is a hyperring with the hyperaddition and multiplication defined by Clearly, B is a (2,2)-absorbing hyperideal of H. In the following, we consider the relationship between a (k,n)-absorbing hyperideal and its radical. Theorem 3.4. Let Hbe a commutative Krasner (m,n)-hyperring. If Iis an (k,n)-absorbing hyperideal of H, then I(m,n)is an (k,n)-absorbing hyperideal and g(x(kn−k+1))∈Ifor all x∈I(m,n). Proof Let x∈I(m,n), that is g(x(t),1g(n−t))∈I for some t≤n or gk(x(t)∈I for t≥n,t=k(n−1)+1. In the first case, using the fact that I is a hyperideal and (k,n)-absorbing, we conclude that g(x(kn−k+1))∈I. In the second case, it is clear. Now, let x1,…,xkn−k+1∈H such that g(x1,x2,…,xkn−k+1)∈I(m,n). Then, there exists t∈Z+ such that g(g(x1,…,xkn−k+1)(t),1g(n−t))∈I,t≤n or g(k)(g(x1,…,xkn−k+1)(t))∈I,t>n,t=k(n−1)+1. Then,   g(g(x1,…,xkn−k+1)(kn−k+1))=g(g(x1(kn−k+1)),…,g(xkn−k+1(kn−k+1)))∈I.Since I is (k,n)-absorbing, it follows that a g-product of (k−1)n−k+2 of the g(xi)(kn−k+1)’s is in I. Thus, a g-product of (k−1)n−k+2 of the   g(x1,…,xi^,…,xkn−k+1)(kn−k+1)∈I,forsomei∈[1,kn−k+1].This implies that g(x1,…,xi^,…,xkn−k+1)∈I(m,n) and we see that I(m,n) is (k,n)-absorbing.□ In the following, we investigate what happens with the intersection of (k,n)-absorbing hyperideals. Theorem 3.5. Let Hbe a Krasner (m,n)-hyperring. If I1,I2are two n-ary prime hyperideals of H, then I1∩I2is a 2-absorbing hyperideal of H. Proof Let x1,x2,…,x2n−1∈H such that g(x1,x2,…,x2n−1)∈I1∩I2. Then, g(x1,x2,…,x2n−1)∈I1 and g(x1,x2,…,x2n−1)∈I2. Since I1 is n-ary prime hyperideal and g(x1,x2,…,g(xn,…,x2n−1))∈I1, then we have either x1∈I1 or x2∈I1 or … or g(xn,…,x2n−1)∈I1 and thus for the same reason xn∈I1 or … or x2n−1∈I1. Similarly, x1∈I2 or x2∈I2 or … or x2n−1∈I2. Let us assume that x1∈I1∩I2. Then, g(x1,x2,…,xn)∈I1∩I2 and the g-product of x1 with n−1 of the xi’s (other than g(x1,x2,…,xn)) is in I1∩I2 since I1∩I2 is a hyperideal. Thus, I1∩I2 is 2-absorbing.□ In the following, we generalize the above result, investigating the structure of the intersection of s hyperideals that are each (kj,n)-absorbing hyperideals of H. Theorem 3.6. Let Hbe a commutative Krasner (m,n)-hyperring. If Ijis an (kj,n)-absorbing hyperideal of Hfor each 1≤j≤s, then ⋂j=1sIjis an (k,n)-absorbing hyperideal, where k=∑j=1skj. Proof Let I1,…,Is be proper hyperideals of H such that Ij is (kj,n)-absorbing and let k0>k1+⋯+ks. Let us suppose that g(x1,…,xk0n−k0+1)∈⋂j=1sIj. Then, for all j, there exists a g-product of (kj−1)n−kj+2 of those k0n−k0+1 elements in Ij, since each Ij is (kj,n)-absorbing. Let the collection of those elements be denoted Aj. Then, let A=⋃j=1k0Aj. Thus, A has at most (k1n−k1+2)+⋯+(ksn−ks+2) elements. Since Ij is a hyperideal, the g-product of all elements of A must be in each Ij. So, ⋂j=1sIj contains a g-product of at most (k1n−k1+2)+⋯+(ksn−ks+2) elements. Thus, the intersections of the Ij’s is an (k1+···+ks,n)-absorbing hyperideal of H.□ Theorem 3.7. Let Hbe a commutative Krasner (m,n)-hyperring. If Iis an (k,n)-absorbing hyperideal of H, then Iis an (t,n)-absorbing hyperideal of Hfor all t≥k. Proof We use the induction on k that if I is (k,n)-absorbing, then it is (k+1,n)-absorbing. Let us suppose that I is 2-absorbing. We will show that I is 3-absorbing. Let x1,x2,…,x3n−2∈H such that g(x1,…,x2n−2,g(x2n−1,…,x3n−2))∈I. Since I is 2-absorbing, it follows that there are n elements of the xi’s (other than g(x2n−1,…,x3n−2)) whose g-product is in I and since I is a hyperideal, then there are 2n−1 elements of the xi’s whose g-product is in I. Thus, I is 3-absorbing. Now, suppose that I is (k,n)-absorbing. We will show that I is (k+1,n)-absorbing. Let   g(g(x1,…,x2n−2),x2n−1,…,x(k+1)n−(k+1)+1)∈I,forx1,…,x(k+1)n−(k+1)+1∈H.Since I is (k,n)-absorbing, it follows that either g(g(x1,…,x2(n−1)),x2n−1,…,xi^,…,x(k+1)n−(k+1)+1)∈I for some i∈[2(n−1),(k+1)n−(k+1)+1] or we have g(x2n−1,…,x(k+1)n−(k+1)+1)∈I. In the first case it is obvious that I is (k+1,n)-absorbing, so we consider only the second case holds, that is g(x2n−1,…,x(k+1)n−(k+1)+1)∈I. Since x1,…,x2(n−1)∈H, then by the definition of hyperideal we have g(x1,…,xn−1,x2n−1,…,x(k+1)n−(k+1)+1)∈I and so, I is (k+1,n)-absorbing.□ Example 3.8. Consider the set of integers Z and its multiplicative subgroup G={1,−1}. The Krasner construction Z/G is a principal hyperideal hyperdomain, that is, it is a hyperring with no zero divisors and whose hyperideals are generated by a single element. Also, the prime hyperideals of Z/G are of the form I=⟨pG⟩ where p is a prime number. Clearly, I is a prime hyperideal (or 1-absorbing) and we observe that I is 2-absorbing. For example, for x,y,z∈Z/G and x(yz)∈I, we have that since I is prime, x∈I or yz∈I, and thus I is (2,2)-absorbing. 4. On (k,n)-absorbing primary hyperideals In his paper [5], Badawi et al. introduced a generalization of primary ideals of commutative rings, which they defined as 2-absorbing primary ideals. We generalize this notion for hyperrings. A proper, nonzero hyperideal I of a commutative Krasner (m,n)-hyperring H is said to be (2,n)-absorbing primary hyperideal of H if whenever x1,…,x2n−1∈H with g(x1,…,x2n−1)∈I, then g(x1,…,xn)∈I or there are n elements of the xi’s (other than g(x1,…,xn)) whose g-product is in I(m,n). Example 4.1. Let us consider the hyperring H=Z12/Z12* of Example 2.2. Let I={0Z12*,4Z12*} be a hyperideal of H. The radical of I is I={0Z12*,2Z12*,4Z12*,6Z12*}. We can observe that I is (2,2)-absorbing primary hyperideal of H. Definition 4.2. Let H be a commutative Krasner (m,n)-hyperring and k a positive integer. A proper hyperideal I of H is said to be a (k,n)-absorbing primary hyperideal of H if whenever x1,…,xkn−k+1∈H and g(x1,…,xkn−k+1)∈I, then either g(x1,x2,…,x(k−1)n−k+2)∈I or a g-product of (k−1)n−k+2 of the xi’s (other than g(x1,x2,…,x(k−1)n−k+2)) is in I(m,n). We see that if there are no g-products of (k−1)n−k+2 elements in I, then we have (k−1)n−k+2 g-products of the form g(x1,…,xkn−k+1)∈I, that is, g(x2,…,xkn−k+1,x1)∈I, g(x3,…,xkn−k+1,x1,x2)∈I, and so forth. Thus, we see that if any two g-products of (k−1)n−k+2 elements are in I(m,n), then there is a g-product of (k−1)n−k+2 elements in I(m,n) for each case. Theorem 4.3. Every n-ary primary hyperideal of a commutative Krasner (m,n)-hyperring is a (2,n)-absorbing primary hyperideal. Proof Let H be a commutative Krasner (m,n)-hyperring and I an n-ary primary hyperideal of H. Then, for g(g(x1,…,xn),…,x2n−1)∈I we have either g(x1,x2,…,xn)∈I or g(xn+1,…,x2n−1)∈I(m,n). Since I(m,n) is a hyperideal and x1,…,xn∈H, we have g(x1,xn+1,…,x2n−1)∈I(m,n) or … or g(xn,xn+1,…,x2n−1)∈I(m,n). Thus, I is a 2-absorbing primary hyperideal.□ Theorem 4.4. Let Hbe a commutative Krasner (m,n)-hyperring. Then, every (k,n)-absorbing primary hyperideal of His a (t,n)-absorbing primary hyperideal for t>n. Proof We show that every (k,n)-absorbing primary hyperideal of H is a (k+1,n)-absorbing primary hyperideal of H. Let I be a (k,n)-absorbing primary hyperideal. Suppose that, Let g(g(x1,…,xn+2),xn+3,…,x(k+1)n−(k+1)+1)∈I for x1,…,x(k+1)n−(k+1)+1∈H. Let g(x1,…,xn+2)=x1′. Since I is a (k,n)-absorbing primary hyperideal of H, it follows that either g(x1′,…,x(k+1)n−(k+1)+1)∈I or a g-product of kn−k+1 of the xi’s (other than g(x1′,…,x(k+1)n−(k+1)+1)) is in I(m,n). Then, since x1,…,xn+2∈H, we have by definition of hyperideal that g(x1,xn+3,…,x(k+1)n−(k+1)+1)∈Ior…org(xn+2,xn+3,…,x(k+1)n−(k+1)+1)∈I. Thus, I is a (k+1,n)- absorbing primary hyperideal of H.□ It is clear that any (k,n)-absorbing hyperideal of H is a (k,n)-absorbing primary hyperideal of H. The converse in general is not true. Example 4.5. Let R=[0,1]. Then, (R,⊕,·) is a Krasner hyperring, where ⊕:R×R→*(R) is the multi-valued function defined by   x⊕y={{max{x,y}}ifx≠y[0,x]ifx=yand · is the usual multiplication on real numbers. Furthermore, let K=[0,0.5]. Then, K is a hyperideal. Clearly, it is not a (2,2)-absorbing hyperideal, while it is a (2,2)-absorbing primary hyperideal of R. Theorem 4.6. Let Hbe a commutative Krasner (m,n)-hyperring. If Iis a (k,n)-absorbing primary hyperideal of H, then I(m,n)is a (k,n)-absorbing hyperideal of H. Proof Let x1,x2,…,xkn−k+1∈H such that g(x1,x2,…,xkn−k+1)∈I(m,n) and suppose that all products of (k−1)n−k+2 of the xi’s except g(x1,x2,…,x(k−1)n−k+2) are not in I(m,n). In other cases we have the requested and there is nothing to prove. We show g(x1,x2,…,x(k−1)n−k+2)∈I(m,n). Since g(x1,x2,…,xkn−k+1)∈I(m,n), it follows that there exists t∈Z+ such that g(g(x1,x2,…,xkn−k+1)(t),1g(n−t))∈I, for t≤n or g(l)(g(x1,x2,…,xkn−k+1)(t))∈I, for t>n,t=l(n−1)+1. Thus, we have for the first case g(g(x1)(t),g(x2)(t),…,g(xkn−k+1)(t),1g(n−t))∈I, for t≤n. Since I is a (k,n)-absorbing primary hyperideal and since none of the g-products of the xi’s are in I(m,n), we obtain that g(g(x1)(t),g(x2)(t),…,g(x(k−1)n−k+2)(t),1g(n−t))=g(g(x1,x2,…,x(k−1)n−k+2)(t),1g(n−t))∈I. That is, g(x1,x2,…,x(k−1)n−k+2)∈I(m,n). Similar for the other case. Thus, I(m,n) is a (k,n)-absorbing hyperideal of H.□ Definition 4.7. Let H be a commutative Krasner (m,n)-hyperring and I a (k,n)-absorbing primary hyperideal of H. Then, P=I(m,n) is a (k,n)-absorbing hyperideal. We say that I is a (P−k,n)-absorbing primary hyperideal of H. Theorem 4.8. Let Hbe a commutative Krasner (m,n)-hyperring and I1,I2,…,Isbe (P−k,n)-absorbing primary hyperideals of Hfor some (k,n)-absorbing hyperideal Pof H. Then, I=⋂i=1sIiis a (P−k,n)-absorbing primary hyperideal of H. Proof Denote P=I=⋂i=1sIi(m,n). Let g(x1,x2,…,xkn−k+1)∈I for some x1,…,xkn−k+1∈H and let us assume that g(x1,x2,…,x(k−1)n−k+2)∉I. Then, we have g(x1,x2,…,x(k−1)n−k+2)∉Ii for some i∈[1,s]. Since every Ii is a (P−k,n)-absorbing primary hyperideal and g(x1,x2,…,x(k−1)n−k+2)∉Ii, then there is a g-product of (k−1)n−k+2 of the xi’s is in Ii(m,n)=P. So, I is a (P−k,n)-absorbing primary hyperideal of H.□ Theorem 4.9. Let Hbe a commutative Krasner (m,n)-hyperring and Ibe a hyperideal of H. If I(m,n)is a (2,n)-absorbing hyperideal of H, then Iis a (3,n)-absorbing primary hyperideal of H. Proof Let x1,…,x3n−2∈H such that g(x1,…,x3n−2)∈I and g(x1,…,x2n−1)∉I. Since g(x1,…,x3n−2)∈I, it follows that g(g(x1,x2n,…,x3n−2),x2,…,x2n−1)∈I⊆I(m,n). Since I(m,n) is (2,n)-absorbing, it follows that either g(g(x1,x2n,…,x3n−2),x2,…,xn)=g(x1,x2,…,xn,x2n,…,x3n−2)∈I(m,n) or g(g(x1,x2n,…,x3n−2),xn+1,…,x2n−1)=g(x1,xn+1,…,x2n−1,x2n,…,x3n−2)∈I(m,n) or g(x2,…,x2n−1)∈I(m,n) and since I(m,n) is a hyperideal, it follows that g(x2,…,x2n)∈I(m,n). Thus, I is a (3,n)-absorbing primary hyperideal of H.□ Theorem 4.10. Let Hbe a commutative Krasner (m,n)-hyperring and Ia hyperideal of H. If I(m,n)is a (k+1,n)-absorbing primary hyperideal of H, then Iis a (k+1,n)-absorbing primary hyperideal of H. Proof Let g(x1,…,x(k+1)n−(k+1)+1)∈I and g(x1,…,xkn−k+1)∉I. Then, g(x1,…,x(k+1)n−(k+1)+1)∈I⊆I(m,n). Therefore, we have   g(x1,…,xkn−k,g(xkn−k+1,…,x(k+1)n−(k+1)+1))∈I(m,n)which is (k+1,n)-absorbing. Let g(xkn−k+1,…,x(k+1)n−(k+1)+1)=x0. Then, g(x1,…,xi^,…,xkn−k,x0)∈I(m,n) for some i∈{0,1,2,…,n} which is a g-product of kn−k+1 elements if i∈[1,n]. Suppose that i=0. Then, g(x1,…,xkn−k)∈I(m,n). Since x(k+1)n−(k+1)+1∈H and I(m,n) is a hyperideal, we have that g(x1,…,xkn−k,x(k+1)n−(k+1)+1)∈I(m,n). Thus, I is a (k+1,n)-absorbing primary hyperideal of H.□ Theorem 4.11. Let H=H1×H2⋯×Hkn−k+1and Jbe a proper nonzero hyperideal of H. If Jis a (k+1,n)-absorbing primary hyperideal of H, then J=I1×⋯×Ikn−k+1for some proper (k,n)-absorbing primary hyperideals I1,…,Ikn−k+1of H1,…,Hkn−k+1and I1≠H1,…,Ik+1≠Hkn−k+1. Proof Let x1,…,xkn−k+1∈H such that g(x1,…,xkn−k+1)∈I1 and suppose by contradiction that I1 is not a (k,n)-absorbing primary hyperideal of H. Define the following elements of H: a1=(x1,1g,…,1g),a2=(x2,1g,…,1g),…,akn−k+1=(xkn−k+1,1g,…,1g),a(k−1)n−k+2=(1g,0,…,0). Then, we have   g(a1,a2,…,a(k−1)n−k+2)=(g(x1,…,xkn−k+1),0,…,0)∈J,g(a1,…,akn−k+1)=(g(x1,…,xkn−k+1),1g,…,1g)∉J,g(a1,ai^,…,a(k−1)n−k+2)=(g(x1,…,xi^,…,x(k−1)n−k+2),0,…,0)∉J(m,n),for some i∈[1,kn−k+1]. This is a contradiction, since J is (k+1,n)-absorbing primary hyperideal. Thus, I1 must be a (k,n)-absorbing primary hyperideal of H. Similarly, we can show that Ii is a (k,n)-absorbing primary hyperideal. This completes the proof.□ Theorem 4.12. Let H,H′be commutative Krasner (m,n)-hyperrings and h:H→H′a homomorphism. Then, If I′is a (k,n)-absorbing primary hyperideal of H′, then h−1(I′)is a (k,n)-absorbing primary hyperideal of H. If his an epiomorphism and Iis a (k,n)-absorbing primary hyperideal of Hcontaining Ker(h), then h(I)is a (k,n)-absorbing primary hyperideal of H′. Proof (1) Let x1,…,xkn−k+1∈H such that g(x1,…,xkn−k+1∈h−1(I′)). We have h(g(x1,…,xkn−k+1))=g(h(x1),…,h(xkn−k+1))∈I′. Since I′ is a (k,n)-absorbing primary hyperideal of H′, it follows that either   g(h(x1),…,h(x(k−1)n−k+2))=h(g(x1,…,x(k−1)n−k+2))∈I′which implies that g(x1,…,x(k−1)n−k+2)∈h−1(I′), or   g(h(x1),…,h(xi)^,…,h(xkn−k+1))=h(g(x1,…,xi^,…,xkn−k+1))∈I′(m,n),which implies g(x1,…,xi^,…,xkn−k+1)∈h−1(I′(m,n))=h−1(I′)(m,n) for some i∈[1,n]. Thus, h−1(I′) is (k,n)-absorbing primary hyperideal. (2) Let x1′,x2′,…,xkn−k+1′∈H′ such that g(x1′,…,xkn−k+1′)∈h(I). Then, there are x1,…,x(k−1)n−k+2∈H such that h(x1)=x1′,…,h(x(k−1)n−k+2)=x(k−1)n−k+2′. We have   h(g(x1,…,xkn−k+1))=g(h(x1),…,h(xkn−k+1))=g(x1′,…,xkn−k+1′)∈h(I).Since Ker(h)⊆I, it follows that g(x1,…,xkn−k+1)∈I. Since I is (k,n)-absorbing primary hyperideal of H, it follows that either g(x1,…,x(k−1)n−k+2)∈I implies h(g(x1,…,x(k−1)n−k+2))=g(h(x1),…,h(x(k−1)n−k+2))=g(x1′,…,x(k−1)n−k+2′)∈h(I) or g(x1,…,xi^,…,xkn−k+1)∈I(m,n) implies h(g(x1,…,xi^,…,xkn−k+1))=g(h(x1),…,h(xi)^,…,h(xkn−k+1))=g(x1′,…,xI′^,…,xkn−k+1′)∈h(I(m,n))⊆h(I)(m,n) for some i∈[1,(k−1)n−k+2]. Thus, h(I) is a (k,n)-absorbing primary hyperideal of H′.□ 5. Future work Definition 5.1. Let H be a commutative Krasner (m,n)-hyperring. A proper hyperideal I of H is a strongly (k,n)-absorbing hyperideal of H if g(I1,…,Ikn−k+1)⊆I for hyperideals I1,…,Ikn−k+1 of H, then there are (k−1)n−k+2 of the Ii’s whose g-product is in I. The following definition extends the notion of strongly (k,n)-absorbing hyperideals to the notion of strongly (k,n)-absorbing primary hyperideals. Definition 5.2. Let H be a commutative Krasner (m,n)-hyperring. A proper nonzero hyperideal I of H is a strongly (k,n)-absorbing primary hyperideal of H if whenever g(I1,…,Ikn−k+1)⊆I for hyperideals I1,…,Ikn−k+1 of H, then either g(I1,…,I(k−1)n−k+2)⊆I or a g-product of (k−1)n−k+2 of the Ii’s (other than g(I1,…,I(k−1)n−k+2) is in I(m,n)). As a future work, we intend to study properties of these notions. References 1 D. F. Anderson and A. Badawi, On (m, n)-closed ideals of commutative rings, J. Algebra Appl.  16 ( 2017), 1750013. [21 pages]. Google Scholar CrossRef Search ADS   2 R. Ameri and M. 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On (k, n)-absorbing hyperideals in Krasner (m, n)-hyperrings

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Abstract

Abstract Krasner Hyperrings are an important class of algebraic hyperstructures which generalize rings further to allow multiple output values for the addition operation. Krasner (m, n)-hyperrings are a generalization of canonical n-ary hypergroups, a generalization of (m, n)-rings, a generalizations of hyperrings in the sense of Krasner and a subclass of the (m, n)-hyperrings. Badawi introduced a generalization of prime ideals called 2-absorbing ideals and this idea is further generalized in a paper by Anderson and Badawi to a concept called n-absorbing ideals. In this paper, we generalize this notion in Krasner (m, n)-hyperrings introducing and studying the notion of (k, n)-absorbing hyperideal. Several properties of them are provided. Also we introduce the notion of (k, n)-absorbing primary hyperideals of H generalizing the analogs results obtained in rings. Several properties of them are provided. 1. Introduction and preliminaries Algebraic structures play a prominent role in mathematics with wide ranging applications in many disciplines such as theoretical physics, computer sciences, control engineering, information sciences and coding theory. Hyperstructures, as a natural extension of classical algebraic structures, in particular hypergroups, were introduced in 1934 by the French mathematician, Marty, at the 8th Congress of Scandinavian Mathematicians [26]. Since then, a lot of papers and several books have been written on this topic. Nowadays, hyperstructures have a lot of applications to several domains of mathematics and computer science (see [6, 34]) and they are studied in many countries of the world. This theory has been subsequently developed by Corsini [7, 8, 9], Mittas [29, 30], Stratigopoulos [32] and by various authors. Basic definitions and propositions about the hyperstructures are found in [7, 9, 13, 16, 34]. Krasner [22] has studied the notion of hyperfields, hyperrings and then some researchers, namely, Davvaz [14, 15, 19], Vougiouklis [33, 34] and others followed him. In a classical algebraic structure, the composition of two elements is an element, while in an algebraic hyperstructure, the composition of two elements is a set. Several books have been written on hyperstructure theory, see [7, 9, 16, 34]. A recent book on hyperstructures [9] points out on their applications in rough set theory, cryptography, codes, automata, probability, geometry, lattices, binary relations, graphs and hypergraphs. Another book [16] is devoted especially to the study of hyperring theory. Several kinds of hyperrings are introduced and analyzed. The volume ends with an outline of applications in chemistry and physics, analyzing several special kinds of hyperstructures: e-hyperstructures and transposition hypergroups. The theory of suitable modified hyperstructures can serve as a mathematical background in the field of quantum communication systems. n-Ary generalizations of algebraic structures are the most natural way for further development and deeper understanding of their fundamental properties. In [18], Davvaz and Vougiouklis introduced the concept of n-ary hypergroups as a generalization of hypergroups in the sense of Marty. Also, we can consider n-ary hypergroups as a nice generalization of n-ary groups. Leoreanu-Fotea and Davvaz in [25] introduced and studied the notion of a partial n-ary hypergroupoid, associated with a binary relation. Some important results, concerning Rosenberg partial hypergroupoids, induced by relations, are generalized to the case of n-ary hypergroupoids. Davvaz et al. in [17] considered a class of algebraic hypersystems which represent a generalization of semigroups, hypersemigroups and n-ary semigroups. Rings are sets that possess algebraic operations of addition and multiplication of elements in the set, encompassing examples such as the integers and rational numbers. Hyperrings generalize rings further to allow multiple output values for the addition operation. In mathematics, introducing more abstract definitions often serves to clarify and consolidate phenomena that arise in a variety of settings. Hyperrings arise naturally in several settings in algebra, including quadratic form theory, number theory, orderings and ordered algebraic structures, tropical geometry, and multiplicative subgroups of fields. Hyperrings are essentially rings with approximately modified axioms. Hyperrings (R,+,·) are of different types introduced by different researchers. The hyperrings have appeared as a new class of algebraic hyperstructures more general than that of hyperfields, introduced by Krasner [22] in the theory of valued fields. Krasner introduced a type of hyperring (R,+,·) where + is a hyperoperation and · is an ordinary binary operation. Such a hyperring is called a Krasner hyperring. In 2007, Davvaz and Leoreanu-Fotea [16] published a book titled Hyperring Theory and Applications. Several authors generalized the study of ordinary rings to the case of where the ring operations are respectively m-ary and n-ary. (m,n)-rings were studied by Crombez [10], Crombez and Timm [11], Dudek [20] and Lee [23]. Davvaz in [12] and also Mirvakili and Davvaz in [27] defined (m,n)-hyperrings and obtained several results in this respect. Moreover, they [28] introduced Krasner (m,n)-hyperrings as a subclass of (m,n)-hyperrings and as a generalization of Krasner hyperrings. The concept of prime ideal, which arises in the theory of rings as a generalization of the concept of prime number in the ring of integers, plays a highly important role in that theory, as might be expected from the central position occupied by the primes in arithmetic. Badawi [4] introduced a generalization of prime ideals called 2-absorbing ideals and this idea is further generalized by Anderson and Badawi [1, 3] to a concept called n-absorbing ideal. In this paper, we generalize this notion in Krasner (m,n)-hyperrings introducing and studying the notion of (k,n)-absorbing hyperideal. Several properties of them are provided. Also we introduce the notion of (k,n)-absorbing primary hyperideal of H generalizing the analogs results obtained in rings (cf. [5]). Several properties of them are provided. 2. Some basic definitions Recall first the basic terms and definitions from the hyperrings theory. The notion of (m,n)-ary hyperring was introduced by Davvaz [12]. Let H be a nonempty set and f:H×H→*(H) be a mapping, where *(H) denotes the set of all nonempty subsets of H. Then, f is called a binary (algebraic) hyperoperation on H. In general, a mapping f:H×H×⋯×H→*(H), where H appears m times, is called an m-ary (algebraic) hyperoperation, and m is called the arity of this hyperoperation. An algebraic system (H,f), where f is an m-ary hyperoperation defined on H, is called an m-ary hypergroupoid or an m-ary hypersystem. Let f be an m-ary hyperoperation on H and A1,A2,…,Am be nonempty subsets of H. We define   f(A1,A2,…,Am)=⋃{f(x1,x2,…,xm)∣xi∈Ai,i=1,2,…,m}. We shall use the following abbreviated notation: the sequence xi,xi+1,…,xj will be denoted by xij. For j<i, xij is the empty symbol. In this convention,   f(x1,…,xi,yi+1,…,yj,zj+1,…,zm)will be written as f(x1i,yi+1j,zj+1m). In the case when yi+1=⋯=yj=y, the last expression will be written in the form f(x1i,y(j−i),zj+1m). Similarly, for subsets A1,A2,…,Am of H we define   f(A1m)=f(A1,A2,…,Am)=⋃{f(x1m)∣xi∈Ai,i=1,…,m}. An m-ary hyperoperation f is called (i,j)- associative if   f(x1i−1,f(xim+i−1),xm+i2m−1)=f(x1j−1,f(xjm+j−1),xm+j2m−1)holds for fixed 1≤i<j≤m and all x1,x2,…,x2m−1∈H. Note that (i,k)-associativity follows from (i,j)- and (j,k)-associativity. If the above condition is satisfied for all i,j∈{1,2,…,m}, then we say that f is associative. The m-ary hyperoperation f is called commutative iff for all x1,…,xm∈H and for all σ∈Sm,f(x1,x2,…,xm)=f(xσ(1),xσ(2),…,xσ(m)). An m-ary hypergroupoid with the associative m-ary hyperoperation is called an m-ary semihypergroup. An m-ary hypergroupoid (H,f) in which the equation b∈f(a1i−1,xi,ai+1m) has a solution xi∈H for every a1i−1,ai+1m,b∈H and 1≤i≤m is called an m-ary quasihypergroup and when (H,f) is an m-ary semihypergroup, (H,f) is called an m-ary hypergroup. Let B be a nonempty subset of H. Then, B is called an m-ary subhypergroup of (H,f), if f(x1m)⊆B for x1m∈B, and the equation b∈f(b1i−1,xi,bi+1m) has a solution xi∈B for every b1i−1,bi+1m,b∈B and 1≤i≤m. An element e∈H is called a scalar neutral element if x=f(e(i−1),x,e(m−i)), for every 1≤i≤m and for every x∈H. An m-ary polygroup [21] is an m-ary hypergroup (P,f) such that the following axioms hold for all 1≤i,j≤m and x,x1m∈P: There exists a unique element 0∈P such that xi=f(0(i−1),x,0(m−i)). There exists a unitary operation—on P such that x∈f(x1m) implies that xi∈f(−xi−1,…,−x1,x,−xn,…,−xi+1). It is clear that every m-ary group with a scalar neutral element is an m-ary polygroup. A canonical m-ary hypergroup [24] is a commutative m-ary polygroup. An element 0 of an m-ary semihypergroup (H,g) is called a zero element if for every x2n∈H, we have g(0,x2n)=g(x2,0,x3n)=⋯=g(x2n,0)=0. If 0 and 0′ are two zero elements, then 0=g(0′,0(m−1))=0′ and so the zero element is unique. Davvaz in [12] and also Mirvakili and Davvaz in [27] defined (m,n)-hyperrings and obtained several results in this respect. Moreover, they introduced Krasner (m,n)-hyperrings as a subclass of (m,n)-hyperrings and as a generalization of Krasner hyperrings, as follows. Definition 2.1. A Krasner (m,n)-hyperring is an algebraic hyperstructure (R,f,g) which satisfies the following axioms: (R,f) is a cannonical m-ary hypergroup. (R,g) is an n-ary semigroup. The n-ary operation g is distributive with respect to the m-ary hyperoperation f, that is, for every a1i−1,ai+1n,x1m∈R,1≤i≤n,   g(a1i−1,f(x1m),ai+1n)=f(g(a1i−1,x1,ai+1n),…,g(a1i−1,xm,ai+1n)). 0 is a zero element (absorbing element) of the n-ary operation g, that is, for every x2n−1∈R, we have g(0,x2n)=g(x2,0,x3n)=⋯=g(x2n,0)=0. It is clear that every Krasner hyperring is a Krasner (2,2)-hyperring. Also, every Krasner (m,0)-hyperring is a cannonical m-ary hyperring and every Krasner (0,n)-hyperring is an n-ary semigroup. Several examples of Krasner (m,n)-hyperrings can be found in [2, 28, 31]. Let S be a nonempty subset of a Krasner (m,n)-hyperring (R,f,g). If (S,f,g) is a Krasner (m,n)-hyperring, then S is called a subhyperring of R. Let I be a nonempty subset of R and 1≤i≤n; we call I an i-hyperideal of R if I is a subhypergroup of the cannonical m-ary hypergroup (R,f), that is, (I,f) is a cannonical n-ary hypergroup. For every x1n∈R,g(x1i−1,I,xi+1n)⊆I.Also, if for every 1≤i≤n, I is an i-hyperideal, then I is called a hyperideal of R. Every hyperideal of R is a subhyperring of R. A hyperideal P of a Krasner (m,n)-hyperring (R,f,g), such that P≠R, is called an n-ary prime hyperideal if for hyperideals U1,…,Un of R, g(U1n)⊆P implies that U1⊆P or U2⊆P or … or Un⊆P [2]. Equivalently, P is an n-ary prime hyperideal of R if for all a1n∈R, g(a1n)∈P implies a1∈P or … or an∈P. A Krasner (m,n)-hyperring (R,f,g) is commutative if (R,g) is a commutative n-ary semigroup. Also, we say that (R,f,g) is with scalar identity if there exists an element 1g such that x=g(x,1g(n−1)) for all x∈R. A Krasner (m,n)-hyperring (R,f,g) is called an n-ary hyperdomain, if R is a commutative Krasner (m,n)-hyperring with scalar identity and g(a1n)=0 implies that a1=0 or a2=0 or … or an=0 for all a1n∈R. Let I be a hyperideal in a commutative Krasner (m,n)-hyperring (R,f,g) with scalar identity. The radical of I, denoted by I(m,n), is the hyperideal ⋂P, where the intersection is taken over all n-ary prime hyperideals P which contain I. If the set of all n-ary hyperideals containing I is empty, then I(m,n) is defined to be R [2]. By the [2, Theorem 4.23], if I is a hyperideal in a commutative Krasner (m,n)-hyperring (R,f,g) with scalar identity 1g, then I(m,n)={r∈R∣g(r(t),1g(n−t))∈I} for t≤n and I(m,n)={r∈R∣g(l)(r(t))∈I} for t>n,t=l(n−1)+1, for some t∈N. Throughout the paper, we will assume that R is a commutative Krasner (m,n)-hyperring with identity element. 3. On (k,n)-absorbing hyperideals Definition 3.1. Let R be a commutative Krasner (m,n)-hyperring and k a positive integer. A proper hyperideal I of R is an (k,n)-absorbing hyperideal of R if whenever g(x1,…,xkn−k+1)∈I for x1,…,xkn−k+1∈H, then there are (k−1)n−k+2 of the xi’s whose g-product is in I. It is clear that for k=1 we take the n-ary prime hyperideal of Krasner (m,n)-hyperring [2]. For k=1 and n=2, we take the classic prime hyperideal of a Krasner hyperring. Example 3.2. Consider Z12={0,1,2,3,4,5,6,7,8,9,10,11} and the multiplicative group of units Z12*={1,5,7,11} of Z12. The Krasner construction Z12/Z12* is a hyperring. The hyperideals I={0Z12*,3Z12*,6Z12*} and J={0Z12*,2Z12*,4Z12*,6Z12*} are (2,2)-absorbing. Example 3.3. Let H={0,1,a} and B={0,a}. Then, (H,+,·) is a hyperring with the hyperaddition and multiplication defined by Clearly, B is a (2,2)-absorbing hyperideal of H. In the following, we consider the relationship between a (k,n)-absorbing hyperideal and its radical. Theorem 3.4. Let Hbe a commutative Krasner (m,n)-hyperring. If Iis an (k,n)-absorbing hyperideal of H, then I(m,n)is an (k,n)-absorbing hyperideal and g(x(kn−k+1))∈Ifor all x∈I(m,n). Proof Let x∈I(m,n), that is g(x(t),1g(n−t))∈I for some t≤n or gk(x(t)∈I for t≥n,t=k(n−1)+1. In the first case, using the fact that I is a hyperideal and (k,n)-absorbing, we conclude that g(x(kn−k+1))∈I. In the second case, it is clear. Now, let x1,…,xkn−k+1∈H such that g(x1,x2,…,xkn−k+1)∈I(m,n). Then, there exists t∈Z+ such that g(g(x1,…,xkn−k+1)(t),1g(n−t))∈I,t≤n or g(k)(g(x1,…,xkn−k+1)(t))∈I,t>n,t=k(n−1)+1. Then,   g(g(x1,…,xkn−k+1)(kn−k+1))=g(g(x1(kn−k+1)),…,g(xkn−k+1(kn−k+1)))∈I.Since I is (k,n)-absorbing, it follows that a g-product of (k−1)n−k+2 of the g(xi)(kn−k+1)’s is in I. Thus, a g-product of (k−1)n−k+2 of the   g(x1,…,xi^,…,xkn−k+1)(kn−k+1)∈I,forsomei∈[1,kn−k+1].This implies that g(x1,…,xi^,…,xkn−k+1)∈I(m,n) and we see that I(m,n) is (k,n)-absorbing.□ In the following, we investigate what happens with the intersection of (k,n)-absorbing hyperideals. Theorem 3.5. Let Hbe a Krasner (m,n)-hyperring. If I1,I2are two n-ary prime hyperideals of H, then I1∩I2is a 2-absorbing hyperideal of H. Proof Let x1,x2,…,x2n−1∈H such that g(x1,x2,…,x2n−1)∈I1∩I2. Then, g(x1,x2,…,x2n−1)∈I1 and g(x1,x2,…,x2n−1)∈I2. Since I1 is n-ary prime hyperideal and g(x1,x2,…,g(xn,…,x2n−1))∈I1, then we have either x1∈I1 or x2∈I1 or … or g(xn,…,x2n−1)∈I1 and thus for the same reason xn∈I1 or … or x2n−1∈I1. Similarly, x1∈I2 or x2∈I2 or … or x2n−1∈I2. Let us assume that x1∈I1∩I2. Then, g(x1,x2,…,xn)∈I1∩I2 and the g-product of x1 with n−1 of the xi’s (other than g(x1,x2,…,xn)) is in I1∩I2 since I1∩I2 is a hyperideal. Thus, I1∩I2 is 2-absorbing.□ In the following, we generalize the above result, investigating the structure of the intersection of s hyperideals that are each (kj,n)-absorbing hyperideals of H. Theorem 3.6. Let Hbe a commutative Krasner (m,n)-hyperring. If Ijis an (kj,n)-absorbing hyperideal of Hfor each 1≤j≤s, then ⋂j=1sIjis an (k,n)-absorbing hyperideal, where k=∑j=1skj. Proof Let I1,…,Is be proper hyperideals of H such that Ij is (kj,n)-absorbing and let k0>k1+⋯+ks. Let us suppose that g(x1,…,xk0n−k0+1)∈⋂j=1sIj. Then, for all j, there exists a g-product of (kj−1)n−kj+2 of those k0n−k0+1 elements in Ij, since each Ij is (kj,n)-absorbing. Let the collection of those elements be denoted Aj. Then, let A=⋃j=1k0Aj. Thus, A has at most (k1n−k1+2)+⋯+(ksn−ks+2) elements. Since Ij is a hyperideal, the g-product of all elements of A must be in each Ij. So, ⋂j=1sIj contains a g-product of at most (k1n−k1+2)+⋯+(ksn−ks+2) elements. Thus, the intersections of the Ij’s is an (k1+···+ks,n)-absorbing hyperideal of H.□ Theorem 3.7. Let Hbe a commutative Krasner (m,n)-hyperring. If Iis an (k,n)-absorbing hyperideal of H, then Iis an (t,n)-absorbing hyperideal of Hfor all t≥k. Proof We use the induction on k that if I is (k,n)-absorbing, then it is (k+1,n)-absorbing. Let us suppose that I is 2-absorbing. We will show that I is 3-absorbing. Let x1,x2,…,x3n−2∈H such that g(x1,…,x2n−2,g(x2n−1,…,x3n−2))∈I. Since I is 2-absorbing, it follows that there are n elements of the xi’s (other than g(x2n−1,…,x3n−2)) whose g-product is in I and since I is a hyperideal, then there are 2n−1 elements of the xi’s whose g-product is in I. Thus, I is 3-absorbing. Now, suppose that I is (k,n)-absorbing. We will show that I is (k+1,n)-absorbing. Let   g(g(x1,…,x2n−2),x2n−1,…,x(k+1)n−(k+1)+1)∈I,forx1,…,x(k+1)n−(k+1)+1∈H.Since I is (k,n)-absorbing, it follows that either g(g(x1,…,x2(n−1)),x2n−1,…,xi^,…,x(k+1)n−(k+1)+1)∈I for some i∈[2(n−1),(k+1)n−(k+1)+1] or we have g(x2n−1,…,x(k+1)n−(k+1)+1)∈I. In the first case it is obvious that I is (k+1,n)-absorbing, so we consider only the second case holds, that is g(x2n−1,…,x(k+1)n−(k+1)+1)∈I. Since x1,…,x2(n−1)∈H, then by the definition of hyperideal we have g(x1,…,xn−1,x2n−1,…,x(k+1)n−(k+1)+1)∈I and so, I is (k+1,n)-absorbing.□ Example 3.8. Consider the set of integers Z and its multiplicative subgroup G={1,−1}. The Krasner construction Z/G is a principal hyperideal hyperdomain, that is, it is a hyperring with no zero divisors and whose hyperideals are generated by a single element. Also, the prime hyperideals of Z/G are of the form I=⟨pG⟩ where p is a prime number. Clearly, I is a prime hyperideal (or 1-absorbing) and we observe that I is 2-absorbing. For example, for x,y,z∈Z/G and x(yz)∈I, we have that since I is prime, x∈I or yz∈I, and thus I is (2,2)-absorbing. 4. On (k,n)-absorbing primary hyperideals In his paper [5], Badawi et al. introduced a generalization of primary ideals of commutative rings, which they defined as 2-absorbing primary ideals. We generalize this notion for hyperrings. A proper, nonzero hyperideal I of a commutative Krasner (m,n)-hyperring H is said to be (2,n)-absorbing primary hyperideal of H if whenever x1,…,x2n−1∈H with g(x1,…,x2n−1)∈I, then g(x1,…,xn)∈I or there are n elements of the xi’s (other than g(x1,…,xn)) whose g-product is in I(m,n). Example 4.1. Let us consider the hyperring H=Z12/Z12* of Example 2.2. Let I={0Z12*,4Z12*} be a hyperideal of H. The radical of I is I={0Z12*,2Z12*,4Z12*,6Z12*}. We can observe that I is (2,2)-absorbing primary hyperideal of H. Definition 4.2. Let H be a commutative Krasner (m,n)-hyperring and k a positive integer. A proper hyperideal I of H is said to be a (k,n)-absorbing primary hyperideal of H if whenever x1,…,xkn−k+1∈H and g(x1,…,xkn−k+1)∈I, then either g(x1,x2,…,x(k−1)n−k+2)∈I or a g-product of (k−1)n−k+2 of the xi’s (other than g(x1,x2,…,x(k−1)n−k+2)) is in I(m,n). We see that if there are no g-products of (k−1)n−k+2 elements in I, then we have (k−1)n−k+2 g-products of the form g(x1,…,xkn−k+1)∈I, that is, g(x2,…,xkn−k+1,x1)∈I, g(x3,…,xkn−k+1,x1,x2)∈I, and so forth. Thus, we see that if any two g-products of (k−1)n−k+2 elements are in I(m,n), then there is a g-product of (k−1)n−k+2 elements in I(m,n) for each case. Theorem 4.3. Every n-ary primary hyperideal of a commutative Krasner (m,n)-hyperring is a (2,n)-absorbing primary hyperideal. Proof Let H be a commutative Krasner (m,n)-hyperring and I an n-ary primary hyperideal of H. Then, for g(g(x1,…,xn),…,x2n−1)∈I we have either g(x1,x2,…,xn)∈I or g(xn+1,…,x2n−1)∈I(m,n). Since I(m,n) is a hyperideal and x1,…,xn∈H, we have g(x1,xn+1,…,x2n−1)∈I(m,n) or … or g(xn,xn+1,…,x2n−1)∈I(m,n). Thus, I is a 2-absorbing primary hyperideal.□ Theorem 4.4. Let Hbe a commutative Krasner (m,n)-hyperring. Then, every (k,n)-absorbing primary hyperideal of His a (t,n)-absorbing primary hyperideal for t>n. Proof We show that every (k,n)-absorbing primary hyperideal of H is a (k+1,n)-absorbing primary hyperideal of H. Let I be a (k,n)-absorbing primary hyperideal. Suppose that, Let g(g(x1,…,xn+2),xn+3,…,x(k+1)n−(k+1)+1)∈I for x1,…,x(k+1)n−(k+1)+1∈H. Let g(x1,…,xn+2)=x1′. Since I is a (k,n)-absorbing primary hyperideal of H, it follows that either g(x1′,…,x(k+1)n−(k+1)+1)∈I or a g-product of kn−k+1 of the xi’s (other than g(x1′,…,x(k+1)n−(k+1)+1)) is in I(m,n). Then, since x1,…,xn+2∈H, we have by definition of hyperideal that g(x1,xn+3,…,x(k+1)n−(k+1)+1)∈Ior…org(xn+2,xn+3,…,x(k+1)n−(k+1)+1)∈I. Thus, I is a (k+1,n)- absorbing primary hyperideal of H.□ It is clear that any (k,n)-absorbing hyperideal of H is a (k,n)-absorbing primary hyperideal of H. The converse in general is not true. Example 4.5. Let R=[0,1]. Then, (R,⊕,·) is a Krasner hyperring, where ⊕:R×R→*(R) is the multi-valued function defined by   x⊕y={{max{x,y}}ifx≠y[0,x]ifx=yand · is the usual multiplication on real numbers. Furthermore, let K=[0,0.5]. Then, K is a hyperideal. Clearly, it is not a (2,2)-absorbing hyperideal, while it is a (2,2)-absorbing primary hyperideal of R. Theorem 4.6. Let Hbe a commutative Krasner (m,n)-hyperring. If Iis a (k,n)-absorbing primary hyperideal of H, then I(m,n)is a (k,n)-absorbing hyperideal of H. Proof Let x1,x2,…,xkn−k+1∈H such that g(x1,x2,…,xkn−k+1)∈I(m,n) and suppose that all products of (k−1)n−k+2 of the xi’s except g(x1,x2,…,x(k−1)n−k+2) are not in I(m,n). In other cases we have the requested and there is nothing to prove. We show g(x1,x2,…,x(k−1)n−k+2)∈I(m,n). Since g(x1,x2,…,xkn−k+1)∈I(m,n), it follows that there exists t∈Z+ such that g(g(x1,x2,…,xkn−k+1)(t),1g(n−t))∈I, for t≤n or g(l)(g(x1,x2,…,xkn−k+1)(t))∈I, for t>n,t=l(n−1)+1. Thus, we have for the first case g(g(x1)(t),g(x2)(t),…,g(xkn−k+1)(t),1g(n−t))∈I, for t≤n. Since I is a (k,n)-absorbing primary hyperideal and since none of the g-products of the xi’s are in I(m,n), we obtain that g(g(x1)(t),g(x2)(t),…,g(x(k−1)n−k+2)(t),1g(n−t))=g(g(x1,x2,…,x(k−1)n−k+2)(t),1g(n−t))∈I. That is, g(x1,x2,…,x(k−1)n−k+2)∈I(m,n). Similar for the other case. Thus, I(m,n) is a (k,n)-absorbing hyperideal of H.□ Definition 4.7. Let H be a commutative Krasner (m,n)-hyperring and I a (k,n)-absorbing primary hyperideal of H. Then, P=I(m,n) is a (k,n)-absorbing hyperideal. We say that I is a (P−k,n)-absorbing primary hyperideal of H. Theorem 4.8. Let Hbe a commutative Krasner (m,n)-hyperring and I1,I2,…,Isbe (P−k,n)-absorbing primary hyperideals of Hfor some (k,n)-absorbing hyperideal Pof H. Then, I=⋂i=1sIiis a (P−k,n)-absorbing primary hyperideal of H. Proof Denote P=I=⋂i=1sIi(m,n). Let g(x1,x2,…,xkn−k+1)∈I for some x1,…,xkn−k+1∈H and let us assume that g(x1,x2,…,x(k−1)n−k+2)∉I. Then, we have g(x1,x2,…,x(k−1)n−k+2)∉Ii for some i∈[1,s]. Since every Ii is a (P−k,n)-absorbing primary hyperideal and g(x1,x2,…,x(k−1)n−k+2)∉Ii, then there is a g-product of (k−1)n−k+2 of the xi’s is in Ii(m,n)=P. So, I is a (P−k,n)-absorbing primary hyperideal of H.□ Theorem 4.9. Let Hbe a commutative Krasner (m,n)-hyperring and Ibe a hyperideal of H. If I(m,n)is a (2,n)-absorbing hyperideal of H, then Iis a (3,n)-absorbing primary hyperideal of H. Proof Let x1,…,x3n−2∈H such that g(x1,…,x3n−2)∈I and g(x1,…,x2n−1)∉I. Since g(x1,…,x3n−2)∈I, it follows that g(g(x1,x2n,…,x3n−2),x2,…,x2n−1)∈I⊆I(m,n). Since I(m,n) is (2,n)-absorbing, it follows that either g(g(x1,x2n,…,x3n−2),x2,…,xn)=g(x1,x2,…,xn,x2n,…,x3n−2)∈I(m,n) or g(g(x1,x2n,…,x3n−2),xn+1,…,x2n−1)=g(x1,xn+1,…,x2n−1,x2n,…,x3n−2)∈I(m,n) or g(x2,…,x2n−1)∈I(m,n) and since I(m,n) is a hyperideal, it follows that g(x2,…,x2n)∈I(m,n). Thus, I is a (3,n)-absorbing primary hyperideal of H.□ Theorem 4.10. Let Hbe a commutative Krasner (m,n)-hyperring and Ia hyperideal of H. If I(m,n)is a (k+1,n)-absorbing primary hyperideal of H, then Iis a (k+1,n)-absorbing primary hyperideal of H. Proof Let g(x1,…,x(k+1)n−(k+1)+1)∈I and g(x1,…,xkn−k+1)∉I. Then, g(x1,…,x(k+1)n−(k+1)+1)∈I⊆I(m,n). Therefore, we have   g(x1,…,xkn−k,g(xkn−k+1,…,x(k+1)n−(k+1)+1))∈I(m,n)which is (k+1,n)-absorbing. Let g(xkn−k+1,…,x(k+1)n−(k+1)+1)=x0. Then, g(x1,…,xi^,…,xkn−k,x0)∈I(m,n) for some i∈{0,1,2,…,n} which is a g-product of kn−k+1 elements if i∈[1,n]. Suppose that i=0. Then, g(x1,…,xkn−k)∈I(m,n). Since x(k+1)n−(k+1)+1∈H and I(m,n) is a hyperideal, we have that g(x1,…,xkn−k,x(k+1)n−(k+1)+1)∈I(m,n). Thus, I is a (k+1,n)-absorbing primary hyperideal of H.□ Theorem 4.11. Let H=H1×H2⋯×Hkn−k+1and Jbe a proper nonzero hyperideal of H. If Jis a (k+1,n)-absorbing primary hyperideal of H, then J=I1×⋯×Ikn−k+1for some proper (k,n)-absorbing primary hyperideals I1,…,Ikn−k+1of H1,…,Hkn−k+1and I1≠H1,…,Ik+1≠Hkn−k+1. Proof Let x1,…,xkn−k+1∈H such that g(x1,…,xkn−k+1)∈I1 and suppose by contradiction that I1 is not a (k,n)-absorbing primary hyperideal of H. Define the following elements of H: a1=(x1,1g,…,1g),a2=(x2,1g,…,1g),…,akn−k+1=(xkn−k+1,1g,…,1g),a(k−1)n−k+2=(1g,0,…,0). Then, we have   g(a1,a2,…,a(k−1)n−k+2)=(g(x1,…,xkn−k+1),0,…,0)∈J,g(a1,…,akn−k+1)=(g(x1,…,xkn−k+1),1g,…,1g)∉J,g(a1,ai^,…,a(k−1)n−k+2)=(g(x1,…,xi^,…,x(k−1)n−k+2),0,…,0)∉J(m,n),for some i∈[1,kn−k+1]. This is a contradiction, since J is (k+1,n)-absorbing primary hyperideal. Thus, I1 must be a (k,n)-absorbing primary hyperideal of H. Similarly, we can show that Ii is a (k,n)-absorbing primary hyperideal. This completes the proof.□ Theorem 4.12. Let H,H′be commutative Krasner (m,n)-hyperrings and h:H→H′a homomorphism. Then, If I′is a (k,n)-absorbing primary hyperideal of H′, then h−1(I′)is a (k,n)-absorbing primary hyperideal of H. If his an epiomorphism and Iis a (k,n)-absorbing primary hyperideal of Hcontaining Ker(h), then h(I)is a (k,n)-absorbing primary hyperideal of H′. Proof (1) Let x1,…,xkn−k+1∈H such that g(x1,…,xkn−k+1∈h−1(I′)). We have h(g(x1,…,xkn−k+1))=g(h(x1),…,h(xkn−k+1))∈I′. Since I′ is a (k,n)-absorbing primary hyperideal of H′, it follows that either   g(h(x1),…,h(x(k−1)n−k+2))=h(g(x1,…,x(k−1)n−k+2))∈I′which implies that g(x1,…,x(k−1)n−k+2)∈h−1(I′), or   g(h(x1),…,h(xi)^,…,h(xkn−k+1))=h(g(x1,…,xi^,…,xkn−k+1))∈I′(m,n),which implies g(x1,…,xi^,…,xkn−k+1)∈h−1(I′(m,n))=h−1(I′)(m,n) for some i∈[1,n]. Thus, h−1(I′) is (k,n)-absorbing primary hyperideal. (2) Let x1′,x2′,…,xkn−k+1′∈H′ such that g(x1′,…,xkn−k+1′)∈h(I). Then, there are x1,…,x(k−1)n−k+2∈H such that h(x1)=x1′,…,h(x(k−1)n−k+2)=x(k−1)n−k+2′. We have   h(g(x1,…,xkn−k+1))=g(h(x1),…,h(xkn−k+1))=g(x1′,…,xkn−k+1′)∈h(I).Since Ker(h)⊆I, it follows that g(x1,…,xkn−k+1)∈I. Since I is (k,n)-absorbing primary hyperideal of H, it follows that either g(x1,…,x(k−1)n−k+2)∈I implies h(g(x1,…,x(k−1)n−k+2))=g(h(x1),…,h(x(k−1)n−k+2))=g(x1′,…,x(k−1)n−k+2′)∈h(I) or g(x1,…,xi^,…,xkn−k+1)∈I(m,n) implies h(g(x1,…,xi^,…,xkn−k+1))=g(h(x1),…,h(xi)^,…,h(xkn−k+1))=g(x1′,…,xI′^,…,xkn−k+1′)∈h(I(m,n))⊆h(I)(m,n) for some i∈[1,(k−1)n−k+2]. Thus, h(I) is a (k,n)-absorbing primary hyperideal of H′.□ 5. Future work Definition 5.1. Let H be a commutative Krasner (m,n)-hyperring. 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