On z-ideals and z-closure operations of semirings, I

Abstract

The aim of this series of papers is to study z-ideals of semirings. In this article, we introduce some distinguished classes of z-ideals of semirings, which include z-prime, z-semiprime, z-irreducible, and z-strongly irreducible ideals and study some of their properties. Using a z-closure operator, we show the equivalence of these classes of ideals with the corresponding z-ideals that are prime, semirprime, irreducible, and strongly irreducible respectively.