Weakly $(m,n)$-closed submodules over commutative rings

Abstract

Let $R$ be a commutative ring with an identity and $M$ be a unitary $R$-module. For positive integers $m$ and $n$, a proper submodule $N$ of $M$ is called an $(m,n)$-closed submodule if for $r\in R$ and $b\in M$, $r^{m}b\in N$ implies either $r^{n}\in(N:_{R}M)$ or $b\in N$ \cite{Haniece4}. The purpose of this paper is to introduce the concept of weakly $(m,n)$-closed submodules as a generalization of $(m,n)$-closed submodules. A proper submodule $N$ of $M$ is called a weakly $(m,n)$-closed submodule if for $r\in R$ and $b\in M$, $0\neq r^{m}b\in N$ implies either $r^{n}\in(N:_{R}M)$ or $b\in N$. Many properties, examples and characterizations of weakly $(m,n)$-closed submodules are introduced, especially in multiplication modules. Moreover, we investigate the behavior of this structure under module homomorphisms, localizations, quotient modules, Cartesian product and idealizations. Finally, we define two kinds of submodules of the amalgamation module along an ideal and investigate conditions under which they are weakly $(m,n)$-closed submodules.