Varieties of *-regular rings

TL;DR

This paper explores the structure of *-regular rings, demonstrating how subdirectly irreducible rings relate to ultraproducts of simple rings and establishing unit-regularity in certain varieties.

Contribution

It provides a structural characterization of subdirectly irreducible *-regular rings and proves unit-regularity for varieties generated by artinian *-regular rings.

Findings

Subdirectly irreducible *-regular rings are homomorphic images of ultraproducts of simple rings.

Unit-regularity holds for the variety generated by artinian *-regular rings.

The paper advances understanding of the algebraic structure of *-regular rings.

Abstract

Given a subdirectly irreducible *-regular ring R, we show that R is a homomorphic image of a regular *-subring of an ultraproduct of the (simple) eRe, e in the minimal ideal of R. Moreover, unit-regularity is shown for every member of the variety generated by artinian *-regular rings (endowed with unit and pseudo-inversion.