On $ϕ$-1-Absorbing Prime Ideals

Abstract

In this paper, we introduce φ-1-absorbing prime ideals in commutative rings. Let R be a commutative ring with a nonzero identity 1 = 0 and φ : I(R) → I(R) ∪ {∅} be a function where I(R) is the set of all ideals of R. A proper ideal I of R is called a φ-1-absorbing prime ideal if for each nonunits x, y, z ∈ R with xyz ∈ I − φ(I), then either xy ∈ I or z ∈ I. In addition to give many properties and characterizations of φ-1-absorbing prime ideals, we also determine rings in which every proper ideal is φ-1-absorbing prime.