An infinite analogue of rings with stable rank one

Abstract

Replacing invertibility with quasi-invertibility in Bass' first stable range condition we discover a new class of rings, the QB-rings. These constitute a considerable enlargement of the class of rings with stable rank one (B-rings), and include examples like the ring of endomorphisms of a vector space over a field F, and the ring of all row- and column- finite matrices over F. We show that the category of QB-rings is stable under the formation of corners, ideals and quotients, as well as matrices and direct limits. We also give necessary and sufficient conditions for an extension of QB-rings to be again a QB-ring, and show that extensions of B-rings often lead to QB-rings. Specializing to the category of exchange rings we characterize the subset of exchange QB-rings as those in which every von Neumann regular element extends to a maximal regular element, i.e. a quasi-invertible element. Finally we show that the C*- algebras that are QB-rings are exactly the extremally rich C*-algebras previously studied by L.G. Brown and the second author.