Generalizations of 2-absorbing and 2-absorbing primary submodules

Authors : Hojjat MOSTAFANASAB, Ünsal TEKİR, Ece YETKİN ÇELİKEL, Emel A. UĞURLU, Gülşen ULUCAK, Ahmad YOUSİFEAN DARANİ

Pages : 1001-1016

View : 15 | Download : 9

Publication Date : 2019-08-08

Article Type : Research Paper

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 $\phi :Sinsert ignore into journalissuearticles values(M);\rightarrow Sinsert ignore into journalissuearticles values(M);\cup \{\emptyset \}$ be a function where $Sinsert ignore into journalissuearticles values(M);$ 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 $abm\notin \phi insert ignore into journalissuearticles values(N);$, then $am\in N$ or $bm\in N$ or $ab\in insert ignore into journalissuearticles values(N:_{R}M);$ 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 $abm\notin \phi insert ignore into journalissuearticles values(N);$, then $am\in M$-$\mathrm{rad}insert ignore into journalissuearticles values(N);$ or $bm\in M$-$\mathrm{rad}insert ignore into journalissuearticles values(N);$ or $ab\in insert ignore into journalissuearticles values(N:_{R}M);$. 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.
Keywords : phi prime submodule, phi primary submodule, 2 absorbing primary submodule, weakly 2 absorbing primary submodule, phi 2 absorbing submodule, phi 2 absorbing primary submodule