Artinian Meadows

TL;DR

This paper introduces Artinian meadows, algebraic structures derived from Artinian rings with total inverses, and shows they decompose into products of local meadows, providing a canonical construction method.

Contribution

It defines Artinian meadows, proves their unique decomposition into local meadows, and offers a canonical construction from unital commutative rings.

Findings

Artinian meadows decompose into products of local meadows.

A canonical construction method from unital rings is provided.

The inverse of zero is an error term absorbing addition.

Abstract

We introduce the notion of Artinian meadow as an algebraic structure constructed from an Artinian ring which is also a common meadow, i.e.\ a commutative and associative structure with two operations (addition and multiplication) with additive and multiplicative identities and for which inverses are total. The inverse of zero in a common meadow is an error term $\mathbb{a}$ which is absorbent for addition. We show that, in analogy with what happens with commutative unital Artinian rings, Artinian meadows decompose as a product of local meadows in an essentially unique way. We also provide a canonical way to construct meadows from unital commutative rings.