On phi-1-absorbing prime ideals

Abstract

In this paper, we introduce phi-1-absorbing prime ideals in commutative rings. Let R be a commutative ring with a nonzero identity 1 not equal 0 and phi : I(R) -> I(R) boolean OR {theta} be a function where I( R) is the set of all ideals of R. A proper ideal I of R is called a phi-1-absorbing prime ideal if for each nonunits x, y, z is an element of R with xyz is an element of I - phi(I), then either xy is an element of I or z is an element of I. In addition to give many properties and characterizations of phi-1-absorbing prime ideals, we also determine rings in which every proper ideal is phi-1-absorbing prime.