Related papers: On 2-absorbing primary submodules of modules over …

We define a new generalization of n-absorbing ideals in commutative rings called n-absorbing I-primary ideals. We investigate some characterizations and properties of such new generalization. If P is an n-absorbing I-primary ideal of R and…

$(1)$ Let $M\subset N$ be a commutative cancellative torsion-free and subintegral extension of monoids. Then we prove that in the case of ring extension $A[M]\subset A[N]$, the two notions, subintegral and weakly subintegral coincide…

The purpose of the present paper is to prove some properties of the strongly irreducible submodules in the arithmetical and Noetherian modules over a commutative ring. The relationship among the families of strongly irreducible submodules,…

Let M be a fixed left R-module. For a left R-module X, we introduce the notion of M-prime (resp. M-semiprime) submodule of X such that in the case M=R, which coincides with prime (resp. semiprime) submodule of X. Other concepts encountered…

This paper introduce and study weakly 1-absorbing prime ideals in commutative rings. Let $A$ be a commutative ring with a nonzero identity $1\neq 0$. A proper ideal $P$ of $A$ is said to be a weakly 1-absorbing prime ideal if for each…

This paper is concerned with S-co-m modules which are a generalization of co-m modules. In section 2, we introduce the S-small and S-essential submodules of a unitary $R$-module $M$ over a commutative ring $R$ with $1\neq 0$ such that S is…

In our recent work, we introduced a generalization of the prime ideal factorization in Dedekind domains for submodules of finitely generated modules over Noetherian rings. In this article, we find conditions for the intersection of two…

Let $R$ be a commutative ring with $1\neq 0$ and $n$ be a fixed positive integer. A proper ideal $I$ of $R$ is said to be an \textit{$n$-OA ideal} if whenever $a_1a_2\cdots a_{n+1}\in I$ for some nonunits $a_1,a_2,\ldots,a_{n+1}\in R$, then…

Let $R$ be a commutative ring with a non-zero identity, $S$ be a multiplicatively closed subset of $R$ and $M$ be a unital $R$-module. In this paper, we define a submodule $N$ of $M$ with $(N:_{R}M)\cap S=\phi$ to be weakly $S$-prime if…

Let R be a semiring. We say that a non-zero subsemimodule S of an R-semimodule M is second if for each a \in R, we have aS = S or aS = 0. The aim of this paper is to study the notion of second subsemimodules of semimodules over commutative…

Let S be an m-system of a ring R, and P a submodule of a right R-module M. This paper, presents the notion of S-prime submodule and provides some properties and equivalent definitions. We define S-multiplication right module, and prove that…

In this paper we prove that if $R$ is a commutative refinement ring and $M$, $N$ are two $R$-modules then, $M\cong N$ if and only if for every maximal ideal $m$ of $R$, $M_m\cong N_m$. We prove if $R$ is a refinement ring, then every…

Let $R$ be a commutative ring. We investigate $R$-modules which can be written as \emph{finite} sums of {\it {second}} $R$-submodules (we call them \emph{second representable}). We provide sufficient conditions for an $R$-module $M$ to be…

Based upon properties of ordinal length, we introduce a new class of modules, the binary modules, and study their endomorphism ring. The nilpotent endomorphisms form a two-sided ideal, and after factoring this out, we get a commutative…

Let $\Omega$ be a group with identity $e$, $\Gamma$ be a $\Omega$-graded commutative ring and $\Im$ a graded $\Gamma$-module. In this article, we introduce the concept of $gr$-$C$-$2^{A}$-secondary submodules and investigate some properties…

Let R be a commutative ring with identity. The concept of second submodule of an R-module (as a dual notion of prime submodules) was introduced and studied by S.Yassemi in 2001. This notion has obtained a great attention by many authors and…

Let R be a commutative ring with identity and N(R) be the set of all nilpotent elements of R. The aim of this paper is to introduce and study the notion of nil-prime ideals as a generalization of prime ideals. We say that a proper ideal P…

Let $G$ be a group, $R$ be a $G$-graded commutative ring with nonzero unity and $GI(R)$ be the set of all graded ideals of $R$. Suppose that $\phi:GI(R)\rightarrow GI(R)\cup\{\emptyset\}$ is a function. In this article, we introduce and…

Let R be a commutative ring with identity and let M be an R-module. The purpose of this paper is to introduce and investigate the submodules of an R-module M which satisfy the dual of Property A, the dual of strong Property A, and the dual…

Let $R$ be a commutative ring with nonzero identity. Let $\mathcal{I}(R)$ be the set of all ideals of $R$ and let $\delta : \mathcal{I}(R)\longrightarrow \mathcal{I}(R)$ be a function. Then $\delta$ is called an expansion function of ideals…