Valuation Semirings

TL;DR

This paper introduces valuation semirings, explores their properties, and characterizes discrete valuation semirings, extending valuation theory to semirings with new structural insights.

Contribution

It defines valuation semirings and discrete valuation semirings, providing characterizations and properties that extend valuation concepts from rings to semirings.

Findings

A multiplicatively cancellative semiring is a valuation semiring iff its ideals are totally ordered.

If the maximal ideal of a valuation semiring is subtractive, then it is integrally closed.

A semiring is a discrete valuation semiring iff it is a multiplicatively cancellative principal ideal semiring with a nonzero maximal ideal.

Abstract

The main scope of this paper is to introduce valuation semirings in general and discrete valuation semirings in particular. In order to do that, first we define valuation maps and investigate them. Then we define valuation semirings with the help of valuation maps and prove that a multiplicatively cancellative semiring is a valuation semiring if and only if its ideals are totally ordered by inclusion. We also prove that if the unique maximal ideal of a valuation semiring is subtractive, then it is integrally closed. We end this paper by introducing discrete valuation semirings and show that a semiring is a discrete valuation semiring if and only if it is a multiplicatively cancellative principal ideal semiring possessing a nonzero unique maximal ideal. We also prove that a discrete valuation semiring is a Gaussian semiring if and only if its unique maximal ideal is subtractive.