Generalizations of 2-absorbing and 2-absorbing primary submodules

Abstract

In this study, we introduce $\phi $-2-absorbing and $\phi $-2-absorbing primary submodules of modules over commutative rings generalizing the concepts of 2-absorbing and 2-absorbing primary submodules. Let $\phi :S(M)\rightarrow S(M)\cup \{\emptyset \}$ be a function where $S(M)$ denotes the set of all submodules of $M$ and $N$ a proper submodule of an $R$-module $M$. We will say that $N$ is a $\phi $-\textit{2-absorbing submodule} of $M$ if whenever $a,b\in R$, $m\in M$ with $abm\in N$ and $abm\notin \phi (N)$, then $am\in N$ or $bm\in N$ or $ab\in (N:_{R}M)$ and $N$ is said to be a $\phi $-2-absorbing primary submodule of $M$ whenever if $a,b\in R$, $m\in M$ with $abm\in N$ and $abm\notin \phi (N)$, then $am\in M$-$\mathrm{rad}(N)$ or $bm\in M$-$\mathrm{rad}(N)$ or $ab\in (N:_{R}M)$. We investigate many properties of these new types of submodules and establish some characterizations for $\phi $-2-absorbing and $\phi $-2-absorbing primary submodules of multiplication modules.