A Study of S-primary Ideals in Commutative Semirings

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, $I=(v{p}^n)$, $n\in \mathbf{N}$ and for some $p\in \mathbf{P}-{\mathbf{P}}_S$ and $v\in R$ such that $(v)\cap S\mathrm{\ne }\mathrm{\varnothing }$, where $\mathbf{P}$ is the set of all irreducible (prime) elements of R and for a multiplicative subset $S⊊R$, the set ${\mathbf{P}}_S$ defined by ${\mathbf{P}}_S=\{p\in \mathbf{P}:(p)\cap S\mathrm{\ne }\mathrm{\varnothing }\}$.