Some remarks on the comparability of ideals in semirings

TL;DR

This paper explores the structure of ideals in semirings, establishing conditions for uniseriality, analyzing linearly ordered prime ideals, and introducing pseudo-valuation semidomains, thereby advancing the understanding of ideal comparability in semiring theory.

Contribution

It provides new characterizations of uniserial semirings, extends valuation concepts to semirings, and introduces pseudo-valuation semidomains, enriching the theory of ideal orderings.

Findings

A semiring is uniserial iff its matrix semiring is uniserial.

Prime ideals are linearly ordered iff certain divisibility conditions hold.

Prime ideals of pseudo-valuation semidomains are linearly ordered.

Abstract

A semiring is uniserial if its ideals are totally ordered by inclusion. First, we show that a semiring $S$ is uniserial if and only if the matrix semiring $M_n(S)$ is uniserial. As a generalization of valuation semirings, we also investigate those semirings whose prime ideals are linearly ordered by inclusion. For example, we prove that the prime ideals of a commutative semiring $S$ are linearly ordered if and only if for each $x,y \in S$, there is a positive integer $n$ such that either $x|y^n$ or $y|x^n$. Then, we introduce and characterize pseudo-valuation semidomains. It is shown that prime ideals of pseudo-valuation semidomains and also of the divided ones are linearly ordered.