On 2-absorbing ideals of commutative semirings

Abstract

In this paper, we investigate 2-absorbing ideals of commutative semirings and prove that if $\mathfrak{a}$ is a nonzero proper ideal of a subtractive valuation semiring $S$ then $\mathfrak{a}$ is a 2-absorbing ideal of $S$ if and only if $\mathfrak{a}=\mathfrak{p}$ or $\mathfrak{a}=\mathfrak{p}^2$ where $\mathfrak{p}=\sqrt\mathfrak{a}$ is a prime ideal of $S$. We also show that each 2-absorbing ideal of a subtractive semiring $S$ is prime if and only if the prime ideals of $S$ are comparable and if $\mathfrak{p}$ is a minimal prime over a 2-absorbing ideal $\mathfrak{a}$, then $\mathfrak{am} = \mathfrak{p}$, where $\mathfrak{m}$ is the unique maximal ideal of $S$.