On $(n-1,n)$-$\phi$-prime ideals in semirings

Abstract

Let $S$ be a commutative semiring and $T(S)$ bethe set of all ideals of $S$. Let $\phi\:T(S)\to T(S)\cup\{\emptyset\}$ be a function. A proper ideal $I$ of a semiring$S$ is called an $(n-1,n)$-$\phi$-prime ideal of $S$ if$a_{1}a_{2}\cdots a_{n}ın I\setminus \phi(I)$,$a_{1},a_{2},\dots,a_{n}ın S$ implies that $a_{1}a_{2}\cdotsa_{i-1}a_{i+1}\cdots a_{n}ın I$ for some $iın \{1,2,\dots,n\}$.In this paper, we prove several results concerning$(n-1,n)$-$\phi$-prime ideals in a commutative semiring $S$ withnon-zero identity connected with those in commutative ring theory.