On classical n -absorbing submodules
- PDF / 269,487 Bytes
- 6 Pages / 595.276 x 790.866 pts Page_size
- 70 Downloads / 219 Views
Arabian Journal of Mathematics
Osama A. Naji
On classical n-absorbing submodules
Received: 4 October 2018 / Accepted: 2 April 2019 © The Author(s) 2019
Abstract Let R a commutative ring with identity and M be a unitary R-module. In this paper, we investigate some properties of n-absorbing submodules of M as a generalization of 2-absorbing submodules. We also define the classical n-absorbing submodule, a proper submodule N of an R-module M is called a classical n-absorbing submodule if whenever a1 a2 . . . an+1 m ∈ N for a1 , a2 , . . . , an+1 ∈ R and m ∈ M, there are n of ai ’s whose product with m is in N . Furthermore, we give some characterizations of n-absorbing and classical n-absorbing submodules under some conditions. Mathematics Subject Classification
13C05 · 13C13 · 13C99
1 Introduction Throughout this paper, we assume that all rings are commutative with 1 = 0. Let R be a commutative ring. An ideal I of R is said to be proper if I = R. Let M a unitary module over R and N be a submodule of M. The residual of N by M, (N : R M) or simply (N : M), denotes the ideal {r ∈ R : r M ⊆ N }. For any element x of M, the ideal (N : x) is defined by (N : x) = {r ∈ R : r x ∈ N }. Let a ∈ R. Then, Na = {x : x ∈ M and ax ∈ N } is a submodule of the R-module M. Let m ∈ M, a cyclic submodule that is generated by m is a submodule of M has the form Rm = {r m : r ∈ R}. A proper submodule N of M is said to be irreducible if N is not an intersection of two submodules of M that properly contain it. The set of zero divisors of M, denoted by Z d(M) is defined by Z d(M) = {r ∈ R : f or some x ∈ M and x = 0, r x = 0}. An R-module M is called a multiplication module if every submodule N of M has the form I M for some ideal I of R. Prime ideals play a crucial role in ring theory, since they interfere with many branches of algebra and they represent an important role in understanding the structure of ring. A proper ideal I of a ring R is called a prime ideal if, whenever ab ∈ I for a, b ∈ R, then a ∈ I or b ∈ I . A proper submodule N of an R-module M is said to be a prime submodule if, whenever a ∈ R, m ∈ M, and am ∈ N , then m ∈ N or a ∈ (N : M). In [5], Badawi introduced a new generalization of prime ideals in a commutative ring R. He defined a nonzero proper ideal I of R to be a 2-absorbing ideal of R if, whenever a, b, c ∈ R and abc ∈ I , then ab ∈ I or ac ∈ I or bc ∈ I . The concept of 2-absorbing ideal has been transferred to modules. A proper submodule N of an R-module M is a 2-absorbing submodule of M [6] if, whenever abm ∈ N for a, b ∈ R and m ∈ M, then am ∈ N or bm ∈ N or ab ∈ (N : M). The class of 2-absorbing submodules of modules was introduced as a generalization of the class of 2-absorbing ideals of rings. Then, many generalizations of 2-absorbing submodules were studied such as primary 2-absorbing [8], almost 2-absorbing [3], almost 2-absorbing primary [2], and classical 2-absorbing [9]. In this article, we investigate some properties of n-absorbing submodules of M as a generalization of 2-absorbing submodules.
Data Loading...
