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Varieties of *-regular rings

Christian Herrmann

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.

Varieties 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.

Paper Structure

This paper contains 6 sections, 6 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Tyukavkin approximation revisited
  4. Simple generators
  5. Unit-regular extensions
  6. Open problems

Key Result

Lemma 1

$S=\{ \alpha \in A\mid \exists a \in R.\: a \sim \alpha\}$ is a $*$-regular $*$-$\Lambda$-subalgebra of $A$ and there is a surjective homomorphism $\varphi:S\to R$ such that $\varphi(\alpha)=a$ if and only if $a \sim \alpha$. Moreover, for the canonical homomorphism $\psi$ from $A$ onto the reduced

Theorems & Definitions (12)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • Lemma 4
  • proof
  • Theorem 5
  • proof
  • ...and 2 more