In this paper, we introduce the concept of 2-absorbing submodules as a generalization of 2-absorbing ideals. Let R be a commutative ring and M an R-module. A proper submodule N of M is called 2-absorbing if whenever a, b ∈ R, m ∈ M and abm ∈ N, then am ∈ N or bm ∈ N or ab ∈ N:RM. Let N be a 2-absorbing submodule of M. It is shown that N:RM is a 2-absorbing ideal of R and either Ass R(M/N) is a totally ordered set or Ass R(M/N) is the union of two totally ordered sets. Furthermore, it is shown that if M is a finitely generated multiplication module over a Noetherian ring R, and Ass R(M/N) a totally ordered set, then N is 2-absorbing whenever N:RM is a 2-absorbing ideal of R. Also, the 2-absorbing avoidance theorem is proved.
Let R be a commutative ring and M be a Noetherian R-module. The intersection graph of annihilator submodules of M , denoted by GA(M) is an undirected simple graph whose vertices are the classes of elements of ZR(M) \ AnnR(M), for a, b ∈ R two distinct classes [a] and [b] are adjacent if and only if AnnM (a) ∩ AnnM (b) = 0. In this paper, we study diameter and girth of GA(M) and characterize all modules that the intersection graph of annihilator submodules are connected. We prove that GA(M) is complete if and only if ZR(M) is an ideal of R. Also, we show that if M is a finitely generated R-module with r(AnnR(M)) = AnnR(M) and |m − AssR(M)| = 1 and GA(M) is a star graph, then r(AnnR(M)) is not a prime ideal of R and |V (GA(M))| = | Min AssR(M)| + 1.
In this paper, we introduce the Krull dimension-dependent elements of a Noetherian commutative ring. Let [Formula: see text] be non-unit elements of a commutative ring [Formula: see text]. [Formula: see text] are called Krull dimension-dependent elements, whenever [Formula: see text] We investigate the elements of a ring according to this property. Among the many results, we characterize the rings that all elements of them are Krull dimension-dependent and we call them, closed under the Krull dimension. Moreover, we determine the structure of the rings with Krull dimension at most 1, that are closed under the Krull dimension.
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