On $z$-ideals and $z$-closure operations of semirings, I
Amartya Goswami
TL;DR
This work extends the theory of $z$-ideals from rings to commutative semirings by introducing a $z$-closure operator on all ideals. It establishes an overarching equivalence: distinguished $z$-ideals (notably $z$-prime, $z$-semiprime, $z$-irreducible, and $z$-strongly irreducible) correspond to their classical counterparts via the $z$-closure, and it develops foundational properties of $z$-ideals, including $z$-radicals and closure under intersections and, under certain conditions, products. The paper then defines and analyzes five key $z$-ideals ($z$-maximal, $z$-prime, $z$-semiprime, $z$-irreducible, $z$-strongly irreducible), proving representation theorems (every $z$-ideal as an intersection of $z$-irreducibles), relationships with prime/semiprime ideals, and structural results in $z$-Noetherian and arithmetical semirings. These results lay a robust groundwork for further topological and structural studies of $ ext{Spec}_{z}(S)$ and hull-kernel frameworks in semiring theory.
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.