Sampad Das | Searcharxiv

arXiv2026

Quasi S-n-ideals in commutative semirings

Let $R$ be a commutative semiring with unity $(1\neq0)$, and let $S$ be a proper multiplicatively closed subset of $R$. In this paper, we introduce the notion of quasi $S$-$n$-ideals in commutative semirings. We the study basic property of quasi $S$-$n$-ideals and establish their relationships with quasi $n$-ideals, $S$-$n$-ideals, $S$-prime ideals, and $S$-primary ideals. We also investigate their behavior under localization. Finally, we determine quasi $S$-$n$-ideals in the quotient polynomial semiring $R[x]/\langle x^m\rangle$ and study their natural extensions via the constructions of idealization and amalgamation.

math.AC↗

arXiv2025

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, $I = (vp^n)$, $n\in \mathbf{N}$ 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 $\mathbf P_S=\{p\in \mathbf P : (p) \cap S \neq \varnothing \}$.

math.AC↗