Modules with local finite bib-ranks and Grothendieck groups of some categories

Abstract

For a commutative semilocal ring R, a remarkable property is that if P is a nitely generated projective R-module with constant rank, then P is free. In this paper, rstly, we extend the above result to the general case (not necessarily commutative). Secondly, we consider the Grothendieck groups of some subcategories of the category of nitely generated projective modules and obtain some exact sequences of K-groups. Finally, we give some applications of the above Grothendieck groups.

Keywords: Semilocal ring, Grothendieck groups, local nite BIB-ranks, local power stably free

Quaestiones Mathematicae 37(2014), 91-109