Prime and primary ideals in characteristic one

TL;DR

This paper investigates the structure of prime and primary ideals in semirings of characteristic one, providing new insights into their properties, counterexamples to primary decomposition, and conditions like Evans' in this setting.

Contribution

It introduces a counterexample to primary decomposition in characteristic one semirings and establishes a weaker form, advancing understanding of ideal theory in this context.

Findings

Counterexample to primary decomposition exists in characteristic one semirings

A weaker version of primary decomposition can be established

Analysis of Evans' condition in semirings of characteristic one

Abstract

We study zero divisors and minimal prime ideals in semirings of characteristic one. Thereafter we find a counterexample to the most obvious version of primary decomposition, but are able to establish a weaker version. Lastly, we study Evans'condition in this context.