On the Subtractive Ideal Structure of Commutative Semirings

TL;DR

This paper explores the structure of ideals in commutative semirings, introducing a subtractive property to restore key ideal-theoretic features and analyzing various classes of semirings.

Contribution

It develops a subtractive analogue of Krull's theorem and investigates ideal properties across different semiring classes, linking ideal theory with order-theoretic concepts.

Findings

Proves a subtractive version of Krull's existence theorem.

Shows the equivalence of $k$-irreducible and $k$-strongly irreducible ideals in arithmetic semirings.

Establishes new characterizations of $k$-prime and $k$-semiprime ideals in additively idempotent semirings.

Abstract

In the theory of commutative semirings, the lack of additive inverses creates a structural divergence between ideals and congruences that does not exist in ring theory. The aim of this article is to restore critical ideal-theoretic properties via the subtractive property. We first prove a subtractive analogue of Krull's existence theorem, guaranteeing the existence of $k$-prime ideals disjoint from multiplicative sets. We show that in arithmetic semirings, the distinction between $k$-irreducible and $k$-strongly irreducible ideals vanishes, a coherence that we show is preserved under localisation. We investigate the structural properties and coincidence phenomena among associated subclasses of $k$-ideals in Laskerian semirings, von Neumann regular semirings, unique factorisation semidomains, principal ideal semidomains, and weakly Noetherian semirings. Finally, within the framework of additively idempotent semirings, we tether subtractive ideal-theoretic structures to underlying order-theoretic constraints, thereby obtaining new characterizations of $k$-prime and $k$-semiprime ideals. In that process, we also establish that every absolutely $k$-prime ideal is $k$-prime and every $k$-maximal ideal is absolutely $k$-prime.