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

For a commutative semilocal ring R, a remarkable property is that if P is a finitely generated projective R-module with constant rank, then P is free. In this paper, firstly, 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 finitely generated projective modules and obtain some exact sequences of K-groups. Finally, we give some applications of the above Grothendieck groups.