On (n−1,n)-ϕ-prime ideals in semirings

Abstract

Let $S$ be a commutative semiring and $T(S)$ bethe set of all ideals of $S$. Let $\varphi T(S)\to T(S)\cup \{\mathrm{\varnothing }\}$ be a function. A proper ideal $I$ of a semiring$S$ is called an $(n-1,n)$-$\varphi$-prime ideal of $S$ if$a_1{a}_2\cdots a_nınI\setminus \varphi (I)$,$a_1,a_2,\dots ,a_nınS$ implies that $a_1{a}_2{\text{cdotsa}}_{i-1}{a}_{i+1}\cdots a_nınI$ for some $iın\{1,2,\dots ,n\}$.In this paper, we prove several results concerning$(n-1,n)$-$\varphi$-prime ideals in a commutative semiring $S$ withnon-zero identity connected with those in commutative ring theory.