Extensions of exchange rings
type: Item
instance of: scholarly article
title: Extensions of exchange rings
zbMATH Open document ID: 0890.16003
author: Pere Ara
published in: Journal of Algebra
publication date: 1998-05-12T00:00:00Z
review text: Given an ideal \(I\) of a ring \(R\), the determination of properties of \(R\) from those of \(I\) and \(R/I\) is known as an `extension problem'. Here the extension problem for exchange rings is solved: \(R\) is an exchange ring if and only if \(I\) and \(R/I\) are exchange rings and idempotents lift modulo \(I\). In the unital case, \(R\) is called an exchange ring provided the regular module \(R_R\) satisfies the (finite) exchange property of \textit{P. Crawley} and \textit{B. Jónsson} [Pac. J. Math. 14, 797-855 (1964; Zbl 0134.25504)] (cf. [\textit{R. B. Warfield}, jun., Math. Ann. 199, 31-36 (1972; Zbl 0238.16024)]). Non-unital exchange rings have not previously occurred, and so the author introduces them and develops their basic properties. With some further work, the extension theorem above is improved as follows: \(R\) is an exchange ring if and only if \(I\) and \(R/I\) are exchange rings and the canonical map \(K_0(R)\to K_0(R/I)\) is surjective. An interesting corollary characterizes semiperfect rings as exactly those semilocal rings \(R\) for which the canonical map \(K_0(R)\to K_0(R/J(R))\) is surjective. These results give completely algebraic generalizations of some extension results concerning C*-algebras of real rank zero, i.e., C*-algebras in which self-adjoint elements can be approximated arbitrarily closely by real linear combinations of orthogonal projections. The connection is a result of the author, \textit{K. C. O'Meara, E. Pardo}, and the reviewer, that a C*-algebra has real rank zero if and only if it is an exchange ring [Separative cancellation for projective modules over exchange rings, Isr. J. Math. (to appear)]. \textit{L. G. Brown} and \textit{G. K. Pedersen} proved [in J. Funct. Anal. 99, No. 1, 131-149 (1991; Zbl 0776.46026)] that if \(I\) is a closed ideal in a C*-algebra \(A\), then \(A\) has real rank zero if and only if \(I\) and \(A/I\) have real rank zero and projections lift modulo \(I\); if and only if \(I\) and \(A/I\) have …
Mathematics Subject Classification ID: 16E50
Mathematics Subject Classification ID: 16E20
Mathematics Subject Classification ID: 46L05
zbMATH DE Number: 1104106
zbMATH Keywords: exchange rings
zbMATH Keywords: extensions
zbMATH Keywords: idempotents
zbMATH Keywords: liftings
zbMATH Keywords: projections
zbMATH Keywords: C*-algebras
zbMATH Keywords: real rank zero
zbMATH Keywords: semiperfect rings
reviewed by: Kenneth R. Goodearl
MaRDI profile type: Publication
full work available at URL: jabr.1997.7116
OpenAlex ID: W2084343509
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DOI: 10.1006/JABR.1997.7116
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description: scientific article; zbMATH DE number 1104106