On regular rings with involution
type: Item
instance of: scholarly article
title: On regular rings with involution
zbMATH Open document ID: 0534.16008
DOI: 10.1007/BF01772930
author: Pere Ara
author: Pere Menal
published in: Archiv der Mathematik
publication date: 1984-01-01T00:00:00Z
review text: This paper is concerned with *-regular rings, i.e., (von Neumann) regular rings with involution in which every principal right ideal is generated by a projection. Handelman has asked whether every *-regular ring is directly finite (that is, \(xy=1\) implies \(yx=1)\). Here the authors prove that every *-regular ring is at least *-finite, that is, \(xx^*=1\) implies \(x^*x=1\). In a related result, they prove that any ring of positive characteristic with a proper involution (that is, \(xx^*=0\) implies \(x=0)\) must be *-finite. The remaining major result of the paper concerns the regular ring of a finite Rickart \(C^*\)-algebra (i.e., a *- finite \(C^*\)-algebra in which the right annihilator of any element is a principal right ideal generated by a projection). \textit{D. Handelman} proved that any finite Rickart \(C^*\)-algebra T is a *-subring of an \(\aleph_ 0\)-continuous *-regular ring Q such that the projections in Q all lie in T [Studies in Analysis, Adv. Math., Suppl. Stud., Vol. 4, 171- 196 (1979; Zbl 0511.46054); Theorem 2.1]. He later showed that if the 2\(\times 2\) matrix algebra \(M_ 2(T)\) is a finite Rickart \(C^*\)- algebra, then T is a classical order in Q, that is, all non-zero-divisors in T are invertible in Q and all elements of Q can be expressed in the forms \(ab^{-1}\) and \(c^{-1}d\) for \(a,b,c,d\in T\) with b and c non- zero-divisors [Adv. Math. 48, 1-15 (1983; Zbl 0524.46037); Theorem 8.3]. This result was used by \textit{D. Handelman}, \textit{J. Lawrence}, and the reviewer to show that if T has no one-dimensional representations, then T is a classical order in Q [Mem. Am. Math. Soc. 234 (1980; Zbl 0435.16005), Theorem III.16.9]. (A simplification of these arguments, due to Berberian, appears in another paper of \textit{D. Handelman} [Rocky Mt. J. Math. 11, 337-381 (1981; 473.16013), Corollary 1.2].) In the present paper, the authors give a very nice short direct proof that T is always a classical order in Q. This is done by showing that…
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 16E50
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46K05
zbMATH DE Number: 3847594
zbMATH Keywords: directly finite rings
zbMATH Keywords: von Neumann regular rings with involution
zbMATH Keywords: *- regular rings
zbMATH Keywords: finite Rickart \(C^*\)-algebra
zbMATH Keywords: *-finite \(C^*\)-algebra
zbMATH Keywords: \(\aleph_ 0\)-continuous *-regular ring
zbMATH Keywords: matrix algebra
zbMATH Keywords: 473.16013
zbMATH Keywords: classical order
zbMATH Keywords: classical ring of quotients
reviewed by: Kenneth R. Goodearl
MaRDI profile type: Publication
cites work: Complemented Modular Lattices
cites work: The regular ring of a finite AW\(^*\)-algebra
cites work: Q4198895
cites work: Affine representations of Grothendieck groups and applications to Rickart 𝐶*-algebras and ℵ₀-continuous regular rings
cites work: The regular ring and the maximal ring of quotients of a finite Baer \(^*\)- ring
cites work: Coordination Applied to Finite Baer ∗ Rings
cites work: Any orthocomplemented complete modular lattice is a continuous geometry
cites work: Q5562323
cites work: On rings of operators
cites work: The Regular Ring and the Maximal Ring of Quotients of a Finite Baer *-Ring
full work available at URL: https://doi.org/10.1007/bf01772930
OpenAlex ID: W2012453851
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description: scientific article; zbMATH DE number 3847594