Exact rings and semirings

TL;DR

This paper introduces the concept of exact semirings, an abstract class characterized by a separation property, to explain the tropical semiring's field-like behavior and unify various algebraic structures.

Contribution

It defines exact semirings, inspired by the tropical semiring, and demonstrates their inclusion of many important rings and semirings, revealing fundamental algebraic properties.

Findings

Exact semirings include many important rings and semirings.

The separation property explains tropical semiring's field-like behavior.

Many algebraic structures share the exactness property.

Abstract

We introduce and study an abstract class of semirings, which we call exact semirings, defined by a Hahn-Banach-type separation property on modules. Our motivation comes from the tropical semiring, and in particular a desire to understand the often surprising extent to which it behaves like a field. The definition of exactness abstracts an elementary property of fields and the tropical semiring, which we believe is fundamental to explaining this similarity. The class of exact semirings turns out to include many other important examples of both rings (proper quotients of principal ideal domains, matrix rings and finite group rings over these and over fields), and semirings (the Boolean semiring, generalisations of the tropical semiring, matrix semirings and group semirings over these).