Rank factorization and bordering of regular matrices over commutative rings

Let R be a commutative ring. Manjunatha Prasad and Bhaskara Rao proved that every regular matrix over R can be completed to an invertible matrix of a particular size by bordering if and only if every regular matrix over R has a rank factorization and if and only if every finitely generated projectiv e R-module is free. Here we consider the case in which the bordering has no prescribed size and in which we take a rank factorization of a suitable ex tension of the given regular matrix. For their prescribed size we discuss t he existence of f is an element of R such that their borderings and their r ank factorizations are true allowing f as a denominator. (C) 2000 Elsevier Science Inc. All rights reserved.