Notes on additively divisible commutative semirings

TL;DR

This paper investigates the structure of additively divisible commutative semirings, revealing conditions under which they are idempotent or contain idempotent ideals, and discusses open questions in the field.

Contribution

It provides new structural results for finitely generated, torsion, and unitless additively divisible commutative semirings, and raises open problems.

Findings

Additively divisible commutative semirings are idempotent if finitely generated and torsion.

One-generated additively divisible semirings without units contain idempotent ideals.

The paper presents open questions about finitely generated commutative semirings.

Abstract

Commutative semirings with divisible additive semigroup are studied. We show that an additively divisible commutative semiring is idempotent, provided that it is finitely generated and torsion. In case that a one-generated additively divisible semiring posseses no unit, it must contain an ideal of idempotent elements. We also present a series of open questions about finitely generated commutative semirings and their equivalent versions.