On Generalization of \(\phi\)-\(n\)-Absorbing Ideals in Commutative Rings

Abstract

The aim of this paper is to extend the concept of \(n\)-absorbing, quasi-\(n\)-absorbing and \(\phi\)-\(n\)-absorbing ideals given by  Anderson and Badawi [2] to the context of \(\phi\)-semi-\(n\)-absorbing ideals. Let \(\phi:\mathcal{I}(R)\rightarrow \mathcal{I}(R)\cup \left\lbrace \emptyset \right \rbrace\) be a function where \(\mathcal{I}(R)\) is the set of all ideals of \(R\). A proper ideal \(R\) of \(R\) is called a \(\phi\)-semi-\(n\)-absorbing Ideal, if for each \(a \in R\) with \(a^{n+1} \in I - \phi(I)\), then \(a^{n}\in I\). Some characterizations of semi-\(n\)-absorbing ideals are obtained.\ It is shown that if \(J\) is a \(\phi\)-semi-\(n\)-absorbing ideal of \(R\), then \(J/I\) is a \(\phi_{I}\)-semi-\(n\)-absorbing ideal of \(R/I\) where \(I \subseteq J\). A number of results concerning relationships between \(\phi\)-semi-\(n\)-absorbing, \(\phi_{0}\)-semi-\(n\)-absorbing, \(\phi_{\emptyset}\)-semi-2-absorbing and \(\phi_{n\geq 2}\)-semi-\(n\)-absorbing ideals of commutative rings. Finally, we obtain sufficient conditions of a semi-\(n\)-absorbing ideal in order to be a \(\phi\)-semi-\(n\)-absorbing ideal.