Further remarks on $z$-ideals of semirings

TL;DR

This paper revisits previous results on $z$-ideals of semirings, removing the need for the bzi-semiring assumption and proposing a more general $z$-radical definition to extend earlier findings.

Contribution

It removes the bzi-semiring assumption and introduces a revised $z$-radical definition for broader applicability of $z$-ideal results.

Findings

Eliminates the need for the bzi-semiring condition.

Provides a more general formulation of $z$-radical.

Extends previous results to a wider class of semirings.

Abstract

In this note, we revisit certain results from a previous work of the second author concerning $z$-ideals of semirings. We demonstrate that the assumption of the semiring being a bzi-semiring can be dispensed with. Additionally, we propose a revised definition of the $z$-radical that enables a more general formulation of results from the earlier work.