The structure of countably generated projective modules over regular rings

We prove that, for every regular ring R, there exists an isomorphism between the monoids of isomorphism classes of finitely generated projective right modules over the rings EndR(R(ω)R) and RCFM(R), where the latter denotes the ring of countably infinite row- and column-finite matrices over R. We use this result to give a precise description of the countably generated projective modules over simple regular rings and over regular rings satisfying s-comparability. © 2000 Academic Press.