Rank factorization and bordering of regular matrices over commutative rings Original Research Article

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 projective 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 extension of the given regular matrix. For their prescribed size we discuss the existence of fset membership, variantR such that their borderings and their rank factorizations are true allowing f as a denominator.