On 2-absorbing ideals of commutative semirings

TL;DR

This paper characterizes 2-absorbing ideals in subtractive valuation semirings, showing their structure relates closely to prime ideals and providing conditions under which they are prime.

Contribution

It provides a complete characterization of 2-absorbing ideals in subtractive valuation semirings and explores their relationship with prime and maximal ideals.

Findings

2-absorbing ideals are either prime or squares of prime ideals

All 2-absorbing ideals are prime if prime ideals are comparable

Minimal primes over 2-absorbing ideals relate to the maximal ideal

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 $S$.