On the Subtractive Ideal Structure of Commutative 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.