Let R be a commutative ring with a nonzero identity and let M be a unitary R-module. We introduce the concepts of 2-absorbing and weakly 2-absorbing submodules of M and give some basic properties of these classes of submodules. Indeed these are generalizations of prime and weak prime submodules. A proper submodule N of M is called a 2-absorbing (resp. weakly 2-absorbing) submodule of M if whenever a, b ∈ R, m ∈ M and abm ∈ N (resp. 0 6= abm ∈ N), then ab ∈ (N :R M) or am ∈ N or bm ∈ N . It is shown that the intersection of each distinct pair of prime (resp. weak prime) submodules of M is 2-absorbing (resp. weakly 2-absorbing). We will also show that if R is a commutative ring, M a cyclic R-module and N a 2-absorbing submodule of M , then either (1) M − radN = P is a prime submodule of M such that P 2 ⊆ N or (2) M − radN = P1 ∩ P2, P1P2 ⊆ N and (M − radN) ⊆ N where P1, P2 are the only distinct minimal prime submodules of N .

2-Absorbing and Weakly 2-Absorbing Submodules

All rings are commutative with $1\neq0$. The purpose of this paper is to investigate the concept of weakly $n$-absorbing ideals generalizing weakly 2-absorbing ideals. We prove that over a $u$-ring $R$ the Anderson-Badawi's conjectures about $n$-absorbing ideals and the Badawi-Yousefian's question about weakly 2-absorbing ideals hold.

On weakly $n$-absorbing ideals of commutative rings

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 AssR(M/N) is a totally ordered set or AssR(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 AssR(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.

On 2-Absorbing Submodules

Abstract Let R be a commutative ring with 1 ≠ 0 {1\\neq 0} . We recall that a proper ideal I of R is called a weakly 2-absorbing primary ideal of R if whenever a , b , c ∈ R {a,b,c\\in R} and 0 ≠ a ⁢ b ⁢ c ∈ I {0\\not=abc\\in I} , then a ⁢ b ∈ I {ab\\in I} or a ⁢ c ∈ I {ac\\in\\sqrt{I}} or b ⁢ c ∈ I {bc\\in\\sqrt{I}} . In this paper, we introduce a new class of ideals that is closely related to the class of weakly 2-absorbing primary ideals. Let I ⁢ ( R ) {I(R)} be the set of all ideals of R and let δ : I ⁢ ( R ) → I ⁢ ( R ) {\\delta:I(R)\\rightarrow I(R)} be a function. Then δ is called an expansion function of ideals of R if whenever L , I , J {L,I,J} are ideals of R with J ⊆ I {J\\subseteq I} , then L ⊆ δ ⁢ ( L ) {L\\subseteq\\delta(L)} and δ ⁢ ( J ) ⊆ δ ⁢ ( I ) {\\delta(J)\\subseteq\\delta(I)} . Let δ be an expansion function of ideals of R. Then a proper ideal I of R (i.e., I ≠ R {I\\not=R} ) is called a weakly 2-absorbing δ-primary ideal if 0 ≠ a ⁢ b ⁢ c ∈ I {0\\not=abc\\in I} implies a ⁢ b ∈ I {ab\\in I} or a ⁢ c ∈ δ ⁢ ( I ) {ac\\in\\delta(I)} or b ⁢ c ∈ δ ⁢ ( I ) {bc\\in\\delta(I)} . For example, let δ : I ⁢ ( R ) → I ⁢ ( R ) {\\delta:I(R)\\rightarrow I(R)} such that δ ⁢ ( I ) = I {\\delta(I)=\\sqrt{I}} . Then δ is an expansion function of ideals of R, and hence a proper ideal I of R is a weakly 2-absorbing primary ideal of R if and only if I is a weakly 2-absorbing δ-primary ideal of R. A number of results concerning weakly 2-absorbing δ-primary ideals and examples of weakly 2-absorbing δ-primary ideals are given.

On weakly 2-absorbing δ-primary ideals of commutative rings

Let R be a commutative ring with 1 6 0. In this paper, we introduce the concept of weakly 2-absorbing primary ideal which is a generalization of weakly 2-absorbing ideal. A proper ideal I of R is called a weakly 2-absorbing primary ideal of R if whenever a,b,c ∈ R and 0 6 abc ∈ I, then ab ∈ I or ac ∈ √ I or bc ∈ √ I. A number of results concerning weakly 2-absorbing primary ideals and examples of weakly 2-absorbing primary ideals are given.

http://www.ndsl.kr/soc_img/society/kms/DBSHBB/2015/v52n1/DBSHBB_2015_v52n1_97.pdf

On weakly 2-absorbing primary ideals of commutative rings

All rings are commutative with identity, and all modules are unital. The pur- pose of this article is to investigate n-absorbing submodules. For this reason we introduce the concept of n-absorbing submodules generalizing n-absorbing ideals of rings. Let M be an R-module. A proper submodule N of M is called an n-absorbing submodule if when- ever a1 ¢¢¢anm 2 N for a1;:::;an 2 R and m 2 M, then either a1 ¢¢¢an 2 (N :R M) or there are n i 1 of ai's whose product with m is in N. We study the basic properties of n-absorbing submodules and then we study n-absorbing submodules of some classes of modules (e.g. Dedekind modules, Prufer modules, etc.) over commutative rings. AMS subject classiflcations: 13A15, 13F05

/pdf/on-n-absorbing-submodules-508e6asy0e.pdf

Fatemeh Soheilnia

Papers

2-Absorbing and Weakly 2-Absorbing Submodules

On weakly $n$-absorbing ideals of commutative rings

On n-absorbing submodules

On weakly 2-absorbing primary submodules of modules over commutative rings