*-Clean Rings; Some Clean and Almost Clean Baer *-rings and von Neumann Algebras

TL;DR

This paper introduces and studies the notions of *-clean and almost *-clean rings, especially in the context of Baer *-rings and von Neumann algebras, establishing new results and open problems in ring theory.

Contribution

It defines *-clean and almost *-clean rings, extends existing theorems to *-rings, and proves that finite type I Baer *-rings and von Neumann algebras are almost *-clean.

Findings

Finite type I Baer *-rings are almost *-clean.

Finite type I von Neumann algebras are almost *-clean.

Several properties are equivalent for certain Baer *-rings, including regularity and *-cleanness.

Abstract

A ring is clean (resp. almost clean) if each of its elements is the sum of a unit (resp. regular element) and an idempotent. In this paper we define the analogous notion for *-rings: a *-ring is *-clean (resp. almost *-clean) if its every element is the sum of a unit (resp. regular element) and a projection. Although *-clean is a stronger notion than clean, for some *-rings we demonstrate that it is more natural to use. The theorem on cleanness of unit-regular rings from [V. P. Camillo, D. Khurana, A Characterization of Unit Regular Rings, Communications in Algebra, 29 (5) (2001) 2293-2295] is modified for *-cleanness of *-regular rings that are abelian (or reduced or Armendariz). Using this result, it is shown that all finite, type I Baer *-rings that satisfy certain axioms (considered in [S. K. Berberian, Baer *-rings, Die Grundlehren der mathematischen Wissenschaften 195, Springer-Verlag, Berlin-Heidelberg-New York, 1972] and [L. Vas, Dimension and Torsion Theories for a Class of Baer *-Rings, Journal of Algebra, 289 (2) (2005) 614-639]) are almost *-clean. In particular, we obtain that all finite type I AW*-algebras (thus all finite type I von Neumann algebras as well) are almost *-clean. We also prove that for a Baer *-ring satisfying the same axioms, the following properties are equivalent: regular, unit-regular, left (right) morphic and left (right) quasi-morphic. If such a ring is finite and type I, it is *-clean. Finally, we present some examples related to group von Neumann algebras and list some open problems.