Covering conditions for ideals in semirings

TL;DR

This paper extends prime avoidance theorems to semirings, introduces the concept of compactly packed semirings, and characterizes their structure and zero-divisors in relation to prime ideals.

Contribution

It generalizes prime avoidance theorems to semiring theory and characterizes compactly packed semirings through prime ideals and zero-divisors.

Findings

Prime avoidance holds for ringoids.

Semirings are compactly packed iff each prime ideal is a radical of a principal ideal.

Zero-divisors are characterized via prime ideals in compactly packed semirings.

Abstract

In this paper, we prove prime avoidance for ringoids. We also generalize McCoy's and Davis' prime avoidance theorems in the context of semiring theory. Next, we proceed to define and characterize compactly packed semirings and show that a commutative semiring is compactly packed if and only if each prime ideal is the radical of a principal ideal. Finally, we calculate the set of zero-divisors of some monoid semimodules over compactly packed semirings in terms of their prime ideals.