1-absorbing primary submodules

Let R be a commutative ring with non-zero identity and M be a unitary R-module. The goal of this paper is to extend the concept of 1-absorbing primary ideals to 1-absorbing primary submodules. A proper submodule N of M is said to be a 1-absorbing primary submodule if whenever non-unit elements a, b ∈ R and m ∈ M with abm ∈ N, then either ab ∈ (N : RM) or m ∈ M − rad(N). Various properties and chacterizations of this class of submodules are considered. Moreover, 1-absorbing primary avoidance theorem is proved.

Graded ϕ-2-Absorbing and Graded ϕ-2-Absorbing Primary Submodules

The main goal of this article is to explore the concepts of graded ϕ-2-absorbing and graded ϕ-2-absorbing primary submodules as a new generalization of the concepts of graded 2-absorbing and graded 2-absorbing primary submodules. Let ϕ:GS(M)→GS(M)⋃{∅} be a function, where GS(M) denotes the collection of graded R-submodules of M. A proper K∈GS(M) is said to be a graded ϕ-2-absorbing R-submodule of M if whenever x,y are homogeneous elements of R and s is a homogeneous element of M with xys∈K−ϕ(K), then xs∈K or ys∈K or xy∈(K:RM), and we call K a graded ϕ-2-absorbing primary R-submodule of M if whenever x,y are homogeneous elements of R and s is a homogeneous element of M with xys∈K−ϕ(K), then xs or ys is in the graded radical of K or xy∈(K:RM). Several properties of these new forms of graded submodules are investigated.

On S-2-absorbing primary submodules

This article introduces the concept of S-2-absorbing primary submodule as a generalization of 2-absorbing primary submodule. Let S be a multiplicatively closed subset of a ring R and M an R-module. A proper submodule N of M is said to be an S-2-absorbing primary submodule of M if (N :R M) ? S = ? and there exists a fixed element s ? S such that whenever abm ? N for some a,b ? R and m ? M, then either sam ? N or sbm ? N or sab ? ?(N :R M). We give several examples, properties and characterizations related to the concept. Moreover, we investigate the conditions that force a submodule to be S-2-absorbing primary.

On graded quasi-primary submodules of graded modules over graded commutative rings

Let $G$ be a group with identity $e$. Let $R$ be a $G$-graded commutative ring and $M$ a graded $R$-module. In this paper, we introduce the concept of graded quasi-primary submodules of graded modules over graded commutative rings. Various properties of graded quasi-primary submodules are considered.

Approximaitly Quasi-primary Submodules

In this paper, we introduce and study the notation of approximaitly quasi-primary submodules of a unitary left -module over a commutative ring with identity. This concept is a generalization of prime and primary submodules, where a proper submodule of an -module is called an approximaitly quasi-primary (for short App-qp) submodule of , if , for , , implies that either or , for some . Many basic properties, examples and characterizations of this concept are introduced.