Vol. 72, No. 2, 1977

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Completions of regular rings. II

Kenneth R. Goodearl

Vol. 72 (1977), No. 2, 423–459

This paper continues earlier investigations into the structure of completions of a (von Neumann) regular ring R with respect to pseudo-rank functions, and into the connections between the ring-theoretic structure of such completions and the geometric structure of the compact convex set P(R) of all pseudo-rank functions on R. In particular, earlier results on the completion of R with respect to a single N P(R) are extended to completions with respect to any nonempty subset X P(R). Completions in this generality are proved to be regular and self-injective by reducing to the case of a single pseudo-rank function, using a theorem that the lattice of σ-convex faces of P(R) forms a complete Boolean algebra. Given a completion R with respect to some X P(R), it is shown that the Boolean algebra of central idempotents of R is naturally isomorphic to the lattice of those σ-convex faces of P(R) which are contained in the σ-convex face generated by X. Consequently, conditions on X are obtained which tell when R is a direct product of simple rings, and how many simple ring direct factors R must have. Also, it is shown that the X-completion of R contains a natural copy of the completion with respect to any subset of X, so in particular the P(R)-completion of R contains copies of all the X-completions of R. The final section investigates the question of when a regular self-injective ring is complete with respect to some family of pseudo-rank functions. It is proved that a regular, right and left self-injective ring R is complete with respect to a family X P(R) provided only that the Boolean algebra of central idempotents of R is complete with respect to X.

Mathematical Subject Classification
Primary: 16A52, 16A52
Received: 28 December 1976
Published: 1 October 1977
Kenneth R. Goodearl
University of California, Santa Barbara
Santa Barbara CA
United States