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.

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.

Approximaitly Primary Submodules

The study deals with the notion of an approximaitly primary submodules of unitary left -module over a commutative ring with identity as a generalization of a primary submodules and approximaitly prime submodules, where a proper submodule of an -module is called an approximaitly primary submodule of , if whenever , for , , implies that either or for some positive integer of . Several characterizations, basic properties of this concept are given. On the other hand the relationships of this concept with some classes of modules are studied. Furthermore, the behavior of approximaitly primary submodule under -homomorphism are discussed http://dx.doi.org/10.25130/tjps.24.2019.098

Algorithm for primary submodule decomposition without producing intermediate redundant components

The aim of this paper is to present an algorithm for the primary decomposition of a submodule [Formula: see text] of [Formula: see text]. We will use for this purpose the algorithms for primary decomposition for ideals in polynomial rings. We will generalize the method of Kawazoe and Noro to primary decomposition for submodules of free modules.

On Generalizations of phi-2-Absorbing Primary Submodules

Let $\phi: S(M) \rightarrow S(M) \cup \left\lbrace \emptyset\right\rbrace $ be a function where $S(M)$ is the set of all submodules of $M$. In this paper, we extend the concept of $\phi$-$2$-absorbing primary submodules to the context of $\phi$-$2$-absorbing semi-primary submodules. A proper submodule $N$ of $M$ is called a $\phi$-$2$-absorbing semi-primary submodule, if for each $m \in M$ and $a_{1}, a_{2}\in R$ with $a_{1}a_{2}m \in N - \phi(N)$, then $a_{1}a_{2}\in \sqrt{(N : M)}$ or $a_{1}m \in N$ or $a^{n}_{2}m\in N$, for some positive integer $n$. Those are extended from $2$-absorbing primary, weakly $2$-absorbing primary, almost $2$-absorbing primary, $\phi_{n}$-$2$-absorbing primary, $\omega$-$2$-absorbing primary and $\phi$-$2$-absorbing primary submodules, respectively. Some characterizations of $2$-absorbing semi-primary, $\phi_{n}$-$2$-absorbing semi-primary and $\phi$-$2$-absorbing semi-primary submodules are obtained. Moreover, we investigate relationships between $2$-absorbing semi-primary, $\phi_{n}$-$2$-absorbing semi-primary and $\phi$-primary submodules of modules over commutative rings. Finally, we obtain necessary and sufficient conditions of a $\phi$-$\phi$-$2$-absorbing semi-primary in order to be a $\phi$-$2$-absorbing semi-primary.

On S-quasi-Dedekind Modules

Let $R$ be a commutative ring and $M$ an unital $R$-module. A proper submodule $L$ of $M$ is called primary submodule of $M$, if $rm\in L$, where $r\in R$, $m\in M$, then $m\in L$ or $r^{n}M\subseteq L$ for some positive integer $n$. A submodule $K$ of $M$ is called semi-small submodule of $M$ if, $K+L\neq M$ for each primary submodule $L$ of $M$. An $R$-module $M$ is called S-quasi-Dedekind module if, for each $f\in End_{R}(M),$ $ f\neq 0$ implies $Kerf$ semi-small in $M$. In this paper we introduce the concept of S-quasi-Dedekind modules as a generalisation of small quasi-Dedekind modules, and gives some of their properties, characterizations and exemples. Another hand we study the relationships of S-quasi-Dedekind modules with some classes of modules and their endomorphism rings.

On 2-absorbing quasi primary submodules

Let R be a commutative ring with nonzero identity, and let M be a nonzero unital R-module. In this article, we introduce the concept of 2-absorbing quasi primary submodules which is a generalization of prime submodules. We define 2-absorbing quasi primary submodule as a proper submodule N of M having the property that abm ? N, then ab ? ?(N :R M) or am ? radM(N) or bm ? radM(N): Various results and examples concerning 2-absorbing quasi primary submodules are given.

On 2-Absorbing Primary Submodules of Modules over Commutative Rings

All rings are commutative with 1 ≠ 0, and all modules are unital. The purpose of this paper is to investigate the concept of 2-absorbing primary submodules generalizing 2-absorbing primary ideals of rings. Let M be an R-module. A proper submodule N of an R-module M is called a 2-absorbing primary submodule of M if whenever a; b ∈ R and m ∈ M and abm ∈ N, then am ∈ M-rad(N) or bm ∈ M-rad(N) or ab ∈(N :R M). It is shown that a proper submodule N of M is a 2-absorbing primary submodule if and only if whenever I1I2K ⊆ N for some ideals I1; I2 of R and some submodule K of M, then I1I2 ⊆ (N :R M) or I1K ⊆ M-rad(N) or I2K ⊆ M-rad(N). We prove that for a submodule N of an R-module M if M-rad(N) is a prime submodule of M, then N is a 2-absorbing primary submodule of M. If N is a 2-absorbing primary submodule of a finitely generated multiplication R-module M, then (N :R M) is a 2-absorbing primary ideal of R and M-rad(N) is a 2-absorbing submodule of M.

On 2-absorbing primary submodules of modules over commutative ring with unity

In this paper, we introduce the concept of a [Formula: see text]-absorbing primary submodule over a commutative ring with nonzero identity which is a generalization of primary submodule. Let [Formula: see text] be an [Formula: see text]-module and [Formula: see text] be a proper submodule of [Formula: see text]. Then [Formula: see text] is said to be a [Formula: see text]-absorbing primary submodule of [Formula: see text] if whenever [Formula: see text], [Formula: see text] and [Formula: see text], then [Formula: see text] or [Formula: see text] or [Formula: see text]. We have given an example and proved number of results concerning [Formula: see text]-absorbing primary submodules. We have also proved the [Formula: see text]-absorbing primary avoidance theorem for submodules.