A Study of S-primary Ideals in Commutative Semirings

TL;DR

This paper introduces and explores the properties of S-primary ideals in commutative semirings, including their structure, decomposition, and localization, extending classical ideal theory to semiring contexts.

Contribution

It defines S-k-irreducible and S-k-maximal ideals, establishes key theorems on S-primary ideals, and characterizes their structure in principal ideal semidomains, advancing semiring ideal theory.

Findings

S-radical of S-primary ideals is prime

Existence and uniqueness theorems for S-primary decompositions

Structure theorem for S-primary ideals in principal ideal semidomains

Abstract

In this article, we define the concept of an $S$-$k$-irreducible ideal and $S$-$k$-maximal ideal in a commutative semiring. We also establish several results concerning $S$-$k$-primary ideals and prove the existence theorem and the $S$-version of the uniqueness theorem using localization, for $S$-$k$-primary decompositions. Also we show that the $S$-radical of every $S$-primary ideal is a prime ideal of $R$. Moreover, we investigate the structure of $S$-primary ideals in principal ideal semidomain and prove that each such ideal can be expressed of the form, $I = (vp^n)$, $n\in \mathbf{N}$ and for some $p \in \mathbf P -\mathbf P_S$ and $v\in R$ such that $(v)\cap S\neq \varnothing $, where $\mathbf P$ is the set of all irreducible (prime) elements of R and for a multiplicative subset $S\subsetneq R$, the set $\mathbf P_S$ defined by $\mathbf P_S=\{p\in \mathbf P : (p) \cap S \neq \varnothing \}$.