Vol. 84, No. 1, 1979

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Homogenization of regular rings of bounded index

Kenneth R. Goodearl and David E. Handelman

Vol. 84 (1979), No. 1, 63–78

This paper is about deriving sufficient conditions for a von Neumann regular ring of bounded index of nilpotence to split into a direct product of homogeneous factors, and with deriving necessary and sufficient conditions for such a ring to be biregular. (Most of the results are actually proved in somewhat greater generality, for regular rings whose primitive factor rings are artinian, or for regular rings which are subdirect products of artinian rings.) The splitting of a regular ring R of bounded index into a direct product of homogeneous factors is obtained from purely numerical hypotheses on the set of lengths of simple artinian factor rings of R. For example, this holds if no element of is a positive integral linear combination of other elements of . In case R is biregular, it suffices to assume that no element of is a multiple of any other element of . Under similar hypotheses on a particular element t ∈ℒ, it is proved that RR1 × R2 such that all simple artinian factor rings of R1 have length t (i.e., R1 is homogeneous of index t) and no simple artinian factor rings of R2 have length t. In the latter part of the paper, topological criteria are derived for R to be biregular. Namely, R is biregular if and only if the prime ideal spectrum of R is Hausdorff, if and only if the set of extreme pseudo-rank functions on R is compact.

Mathematical Subject Classification
Primary: 16A30, 16A30
Received: 21 August 1978
Revised: 26 February 1979
Published: 1 September 1979
Kenneth R. Goodearl
University of California, Santa Barbara
Santa Barbara CA
United States
David E. Handelman