An Infinite Analogue of Rings with Stable Rank One

✍ Scribed by Pere Ara; Gert K Pedersen; Francesc Perera

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 297 KB
• Volume: 230
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.2000.8330

✦ Synopsis

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 Ž . enlargement of the class of rings with stable rank one B-rings and include

Ž . examples like End V , the ring of endomorphisms of a vector space V over some ‫ކ‬ Ž . field ‫,ކ‬ and ‫ނ‬ ‫ކ‬ , the ring of all row-and column-finite matrices over ‫.ކ‬ 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 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 1 Ž .