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

TL;DR

This paper introduces and studies various classes of $z$-ideals in semirings, establishing their properties and equivalences with certain $z$-closure-based ideal classes, advancing the understanding of ideal theory in semirings.

Contribution

It defines new classes of $z$-ideals in semirings and proves their equivalence with existing ideal classes using a $z$-closure operator.

Findings

Defined $z$-prime, $z$-semiprime, $z$-irreducible, and $z$-strongly irreducible ideals.

Proved the equivalence of these classes with prime, semiprime, irreducible, and strongly irreducible $z$-ideals.

Established properties of these $z$-ideals in the context of semirings.

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.