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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Coordinatization applied to finite Baer * rings

Author: David Handelman
Journal: Trans. Amer. Math. Soc. 235 (1978), 1-34
MSC: Primary 16A28
MathSciNet review: 0463230
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Abstract: We clarify and algebraicize the construction of the 'regular rings' of finite Baer $ ^\ast$ rings. We first determine necessary and sufficient conditions of a finite Baer $ ^\ast$ ring so that its maximal ring of right quotients is the 'regular ring', coordinating the projection lattice. This is applied to yield significant improvements on previously known results: If R is a finite Baer $ ^\ast$ ring with right projections $ ^\ast$-equivalent to left projections $ ({\text{LP}} \sim {\text{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 $ ^\ast$ rings, and they also satisfy $ {\text{LP}} \sim {\text{RP}}$; if R is a real $ A{W^\ast}$ algebra without central abelian projections, then all matrix rings over R are also $ A{W^\ast}$.

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 Baer $ ^\ast$ 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 $ ^\ast$ ring satisfying the 'square root' (SR) axiom, and either of type II or possessing 4 or more equivalent projections as above, satisfies $ {\text{LP}} \sim {\text{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.

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Keywords: Finite Baer $ ^\ast$ ring, epimorphism of ring, maximal ring of quotients, complete $ ^\ast$-regular ring, complemented modular lattice, continuous geometry
Article copyright: © Copyright 1978 American Mathematical Society