On Generalization of ϕ-n-Absorbing Ideals in Commutative Rings

Abstract

The aim of this paper is to extend the concept of $n$-absorbing, quasi-$n$-absorbing and $\varphi$-$n$-absorbing ideals given by  Anderson and Badawi [2] to the context of $\varphi$-semi-$n$-absorbing ideals. Let $\varphi :\mathcal{I}(R)\to \mathcal{I}(R)\cup \left\{\mathrm{\varnothing }\right\}$ be a function where $\mathcal{I}(R)$ is the set of all ideals of $R$. A proper ideal $R$ of $R$ is called a $\varphi$-semi-$n$-absorbing Ideal, if for each $a\in R$ with $a^{n+1}\in I-\varphi (I)$, then $a^n\in I$. Some characterizations of semi-$n$-absorbing ideals are obtained.\ It is shown that if $J$ is a $\varphi$-semi-$n$-absorbing ideal of $R$, then $J/I$ is a ${\varphi }_I$-semi-$n$-absorbing ideal of $R/I$ where $I\subseteq J$. A number of results concerning relationships between $\varphi$-semi-$n$-absorbing, ${\varphi }_0$-semi-$n$-absorbing, ${\varphi }_{\mathrm{\varnothing }}$-semi-2-absorbing and ${\varphi }_{n\ge 2}$-semi-$n$-absorbing ideals of commutative rings. Finally, we obtain sufficient conditions of a semi-$n$-absorbing ideal in order to be a $\varphi$-semi-$n$-absorbing ideal.