A completion of reduced commutative rings

TL;DR

This paper introduces a canonical completion of reduced commutative rings, fixing their structural deficiencies by adjoining weak inverses and roots, and characterizes dominions in these classes.

Contribution

It constructs a discriminator variety of reduced rings with regular monomorphisms and provides a simple description of dominions in these classes.

Findings

All monomorphisms become regular in the completion.

The class forms a discriminator variety.

Dominions are explicitly characterized in these classes.

Abstract

A commutative ring is reduced when it can be embedded into a direct product of fields. While the category of reduced commutative rings plays a fundamental role in affine geometry, it exhibits several structural deficiencies: it admits nonregular monomorphisms and epimorphisms, lacks amalgamation, and is not equationally axiomatizable. In this paper, we simultaneously repair these defects via a canonical completion in which all monomorphisms become regular. This completion is obtained by adjoining weak inverses and weak prime roots, turning the class of reduced commutative rings into a discriminator variety. As a consequence, we obtain an explicit description of dominions in every class of reduced commutative rings containing all fields. This description is strikingly simple compared to that of dominions in the category of all commutative rings, as reflected in the Isbell-Mazet-Silver Zigzag Theorem.