Generalizations of 2-absorbing Primal Ideals in Commutative Rings

Let R be a commutative ring with unity (1 =6 0). A proper ideal of R is an ideal I of R such that I =6 R . Let phi : a ( R ) a ( R ) U {O} be any function, where a ( R ) denotes the set of all proper ideals of R . In this paper we introduce the concept of a phi -2-absorbing primal ideal which is a generalization of a phi -primal ideal. An element a E R is defined to be phi -2-absorbing prime to I if for any r, s, t E R with rsta E I \ phi ( I ), then rs E I or rt E I or st E I . An element a E R is not phi -2-absorbing prime to I if there exist r, s, t E R , with rsta E I \ phi ( I ), such that rs, rt, st E R \ I . We denote by nu d, ( I ) the set of all elements in R that are not phi -2-absorbing prime to I . We define a proper ideal I of R to be a phi -2-absorbing primal if the set nu d, ( I ) U phi ( I ) forms an ideal of R . Many results concerning phi -2-absorbing primal ideals and examples of phi -2-absorbing primal ideals are given.