Quasi S-n-ideals in commutative semirings

Let $R$ be a commutative semiring with unity $(1\mathrm{\ne }0)$, 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 $R[x]\mathrm{/}⟨x^m⟩$ and study their natural extensions via the constructions of idealization and amalgamation.