Generalizations of 2-absorbing primal ideals in commutative rings

Abstract

Let $R$ be a commutative ring with unity $(1\not=0)$. A proper ideal of $R$ is an ideal $I$ of $R$ such that $I\not=R$. Let $\phi : \mathfrak{I}(R)\rightarrow\mathfrak{I}(R)\cup\{\emptyset\}$ be any function, where $\mathfrak{I}(R)$ denotes the set of all proper ideals of $R$. In this paper we introduce the concept of a $\phi$-2-absorbing primal ideal which is a generalization of a $\phi$-primal ideal. An element $a\in R$ is defined to be $\phi$-2-absorbing prime to $I$ if for any $r,s,t\in R$ with $rsta\in I\setminus\phi(I)$, then $rs\in I$ or $rt\in I$ or $st\in I$. An element $a\in R$ is not $\phi$-2-absorbing prime to $I$ if there exist $r,s,t\in R$, with $rsta\in I\setminus\phi(I)$, such that $rs, rt, st\in R\setminus I$. We denote by $\nu_\phi(I)$ the set of all elements in $R$ that are not $\phi$-2-absorbing prime to $I$. We define a proper ideal $I$ of $R$ to be a $\phi$-2-absorbing primal if the set $\nu_\phi(I)\cup\phi(I)$ forms an ideal of $R$. Many results concerning $\phi$-2-absorbing primal ideals and examples of $\phi$-2-absorbing primal ideals are given.