GENERALIZATIONS OF PRIMAL IDEALS OVER COMMUTATIVE SEMIRINGS

In this article we generalize some definitions and results from ideals in rings to ideals in semirings. Let R be a commutative semiring with identity. Let phi: v(R) -> v(R) boolean OR {empty set} be a function, where v(R) denotes the set of all ideals of R. A proper ideal I is an element of v(R) is called phi-prime ideal if ra is an element of I - phi(I) implies r is an element of I or a is an element of I. 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 p(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 = p(I) boolean OR phi(I) forms an ideal of R. Throughout this work, we define almost primal and phi-primal ideals, and we also show that they enjoy many of the properties of primal ideals.