On Generalization of \(φ\)-\(n\)-Absorbing Ideals in Commutative Rings - Yiarayong, Pairote; Siripitukdet, Manoj (Communications in Mathematics and Applications, 2018)
The aim of this paper is to extend the concept of \(n\)-absorbing, quasi-\(n\)-absorbing and \(φ\)-\(n\)-absorbing ideals given by Anderson and Badawi [2] to the context of \(φ\)-semi-\(n\)-absorbing ideals. Let \(φ:I(R)→ I(R)brace\) be a function where \(I(R)\) is the set of all ideals of \(R\). A proper ideal \(R\) of \(R\) is called a \(φ\)-semi-\(n\)-absorbing Ideal, if for each \(a R\) with \(a^n+1 I - φ(I)\), then \(a^nI\). Some characterizations of semi-\(n\)-absorbing ideals are obtained. It is shown that if \(J\) is a \(φ\)-semi-\(n\)-absorbing ideal of \(R\), then \(J/I\) is a \(φ_I\)-semi-\(n\)-absorbing ideal of \(R/I\) where \(I J\). A number of results concerning relationships between \(φ\)-semi-\(n\)-absorbing, \(φ_0\)-semi-\(n\)-absorbing, \(φ_\)-semi-2-absorbing and \(φ_n≥ 2\)-semi-\(n\)-absorbing ideals of commutative rings. Finally, we obtain sufficient conditions of a semi-\(n\)-absorbing ideal in order to be a \(φ\)-semi-\(n\)-absorbing ideal.