Prime ideals in the Boolean polynomial semiring

TL;DR

This paper classifies prime ideals in the Boolean polynomial semiring with one variable, disproving a previous conjecture and grouping these ideals into three classes indexed by integers.

Contribution

It provides a complete classification of prime ideals in the Boolean polynomial semiring, correcting prior conjectures and organizing the ideals into three distinct classes.

Findings

Disproved a conjecture of Alarcón and Anderson.

Classified prime ideals into three classes indexed by integers.

Provided a complete understanding of prime ideals in this semiring.

Abstract

In this article, we disprove a conjecture of F. Alarc\'on and D. Anderson and give a complete classification of the prime ideals in the one variable polynomial semiring with coefficients in Boolean semifield. We group the prime ideals of $\mathbb{B}[x]$ into three classes, indexed by integers.