Coordinatization applied to finite Baer *\ rings

Abstract

We clarify and algebraicize the construction of the 'regular rings' of finite Baer * rings. We first determine necessary and sufficient conditions of a finite Baer * ring so that its maximal ring of right quotients is the 'regular ring', coordinatizing the projection lattice. This is applied to yield significant improvements on previously known results: If R is a finite Baer * ring with right projections *-equivalent to left projections (LP RP), and is either of type II or has 4 or more equivalent orthogonal projections adding to 1, then all matrix rings over R are finite Baer * rings, and they also satisfy LP RP; if R is a real A W* algebra without central abelian projections, then all matrix rings over R are also A W*. An alternate approach to the construction of the 'regular ring' is via the Coordinatization Theorem of von Neumann. This is discussed, and it is shown that if a Baet * ring without central abelian projections has a 'regular ring', the 'regular ring' must be the maximal ring of quotients. The following result comes out of this approach: A finite Baer * ring satisfying the 'square root' (SR) axiom, and either of type II or possessing 4 or more equivalent projections as above, satisfies LP RP, and so the results above apply. We employ some recent results of J. Lambek on epimorphisms of rings. Some incidental theorems about the existence of faithful epimorphic regular extensions of semihereditary rings also come out. This work arose from two sources: frequent profitable discussions with Professor J. La m bek, and the comment on p. 212 of [2], which asks if there is any connection between a Baer * ring whose lattice of projections is modular and the regular ring coordinatizing the lattice (the answer is Yes!). It is with the greatest pleasure that I acknowledge my debt to Professor J. Lamabek, my advisor. Letters and preprints from Professor S. K. Berberian, and an interesting conversation with Professor G. Michler helped me clarify many ideas and results. The terminology is that of [13] and [14]. It is assumed the reader has some familiarity with all of the following: Baer * rings [13], the maximal or complete ring of quotients [14, p. 94 on ], (von Neumann) regular rings, and flat epimorphisms of rings [21]. Received by the editors November 10, 1975 and, in revised form, March 12, 1976. (')Supported by a Postdoctoral Fellowship from the National Research Council of Canada. AMS (MOS) subject classifications (1970). Primary 46K99; Secondary 06A30, 06A25, 16A08, 16A28, 16A34, 16A52, 16A80, 46L10.