GENERALIZATIONS OF PRIMAL IDEALS IN COMMUTATIVE RINGS

Let R be a commutative ring with identity. Let empty set: J (R) -> J (R) boolean OR {empty set} be a function where J (R) denotes the set of all ideals of R. Let I be an ideal of R. An element a is an element of R is called phi-prime to I if ra is an element of I -phi (I) (with r is an element of R) implies that r is an element of I. We denote by S-phi (I) the set of all elements of R that are not phi-prime to I. I is called a phi-primal ideal of R if the set P := S-phi(I) boolean OR phi(I) forms an ideal of R. So if we take phi(empty set)(Q) =empty set (resp., phi(0)(Q) =0 a phi-primal ideal is primal (resp., weakly primal). In this paper we study the properties of several generalizations of primal ideals of R.