Rank functions and K 0 of regular rings

Abstract

This paper is concerned with the existence and uniqueness of rank and pseudo-rank functions on a von Neumann regular ring R. The main technique used involves transferring hypotheses becomes a partially ordered abelian group. It is shown that the existence of a pseudo-rank function on R is equivalent to certain finiteness conditions on the matrix rings over R. As a corollary, necessary and sufficient conditions are obtained for the existence of a rank function on a simple regular ring. Uniqueness of a rank function is shown to be equivalent to certain comparability conditions on the principal right ideals of R. Other results concern the existence of enough pseudo-rank functions to distinguish nonzero ring elements from zero, or to distinguish between non-isomorphic principal right ideals. All rings in this paper are associative with unit (but usually noncommutative), and all modules are unital right modules. We use “regular” to mean “von Neumann regular”. The research of the first author was partially supported by National Science Foundation Grant No. GP-43029.