On Generalizations of phi-2-Absorbing Primary Submodules

Abstract

Let $\varphi :S(M)\to S(M)\cup \left\{\mathrm{\varnothing }\right\}$ be a function where $S(M)$ is the set of all submodules of $M$. In this paper, we extend the concept of $\varphi$-$2$-absorbing primary submodules to the context of $\varphi$-$2$-absorbing semi-primary submodules. A proper submodule $N$ of $M$ is called a $\varphi$-$2$-absorbing semi-primary submodule, if for each $m\in M$ and $a_1,a_2\in R$ with $a_1{a}_{2}m\in N-\varphi (N)$, then $a_1{a}_2\in \sqrt{(N:M)}$ or  $a_{1}m\in N$ or $a_2^{n}m\in N$, for some positive integer $n$. Those are extended from $2$-absorbing primary, weakly $2$-absorbing primary, almost $2$-absorbing primary, ${\varphi }_n$-$2$-absorbing primary, $\omega$-$2$-absorbing primary and $\varphi$-$2$-absorbing primary submodules, respectively. Some characterizations of $2$-absorbing semi-primary, ${\varphi }_n$-$2$-absorbing semi-primary and $\varphi$-$2$-absorbing semi-primary submodules are obtained. Moreover, we investigate relationships between $2$-absorbing semi-primary, ${\varphi }_n$-$2$-absorbing semi-primary and $\varphi$-primary submodules of modules over commutative rings. Finally, we obtain necessary and sufficient conditions of a $\varphi$-$\varphi$-$2$-absorbing semi-primary in order to be a $\varphi$-$2$-absorbing semi-primary.