In algebra, the fundamental theorem of algebraic K-theory describes the effects of changing the ring of K-groups from a ring R to or . The theorem was first proved by Bass for and was later extended to higher K-groups by Quillen. Let be the algebraic K-theory of the category of finitely generated modules over a noetherian ring R; explicitly, we can take , where is given by Quillen's Q-construction. If R is a regular ring (i.e., has finite global dimension), then For a noetherian ring R, the fundamental theorem states: * (i) . * (ii) . ); this is the version proved in Grayson's paper.