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 ℒ of lengths of simple artinian factor rings of
R. For example, this holds if no element of ℒ is a positive integral linear
combination of other elements of ℒ. In case R is biregular, it suffices to
assume that no element of ℒ is a multiple of any other element of ℒ. Under
similar hypotheses on a particular element t ∈ℒ, it is proved that R≅R1× 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.
