Generalizations of 2-absorbing primal ideals in commutative rings

Let R be a commutative ring with unity (1=0). A proper ideal of R is an ideal I of R such that I=R. Let φ : I(R)(R)∪\∅\ be any function, where I(R) denotes the set of all proper ideals of R. In this paper we introduce the concept of a φ-2-absorbing primal ideal which is a generalization of a φ-primal ideal. An element a∈ R is defined to be φ-2-absorbing prime to I if for any r,s,t∈ R with rsta∈ Iφ(I), then rs∈ I or rt∈ I or st∈ I. An element a∈ R is not φ-2-absorbing prime to I if there exist r,s,t∈ R, with rsta∈ Iφ(I), such that rs, rt, st∈ R I. We denote by ν_φ(I) the set of all elements in R that are not φ-2-absorbing prime to I. We define a proper ideal I of R to be a φ-2-absorbing primal if the set ν_φ(I)∪φ(I) forms an ideal of R. Many results concerning φ-2-absorbing primal ideals and examples of φ-2-absorbing primal ideals are given.