Skew Meadows

TL;DR

This paper introduces skew meadows, a class of non-commutative rings with an inverse operator, and establishes their relationship with skew fields, including an embedding theorem and a completeness result for their equational logic.

Contribution

It defines skew meadows, explores their properties, and proves an embedding of skew meadows into products of skew fields, linking ring regularity to skew meadow structure.

Findings

All skew fields are skew meadows

Skew meadows can be embedded into products of skew fields

A completeness theorem for the equational logic of skew fields

Abstract

A skew meadow is a non-commutative ring with an inverse operator satisfying two special equations and in which the inverse of zero is zero. All skew fields and products of skew fields can be viewed as skew meadows. Conversely, we give an embedding of non-trivial skew meadows into products of skew fields, from which a completeness result for the equational logic of skew fields is derived. The relationship between regularity conditions on rings and skew meadows is investigated.