On $2$-Absorbing Semiprimary Submodules

Abstract

In this paper, we introduce the concept of a $2$-absorbing semiprimary submodule over a commutative ring with nonzero identity which is a generalization of $2$-absorbing primary submodule. Let $N$ be a proper submodule of an $R$-module $M$. Then $N$ is said to be a$2$-absorbing semiprimary submodule of $M$ if whenever $a_{1}a_{2} \in R, m \in M$ and $a_{1}a_{2}m \in N$, 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 $n$. We have given an example and proved number of results concerning $2$-absorbing semiprimary submodules.