Morita Equivalence - Significance in K-theory

Significance in K-theory

If two rings are Morita equivalent, there is an induced equivalence of the respective categories of projective modules since the Morita equivalences will preserve exact sequences (and hence projective modules). Since the algebraic K-theory of a ring is defined (in Quillen's approach) in terms of the homotopy groups of the classifying space of the nerve of the category of projective modules over the ring, Morita equivalent rings must have isomorphic K-groups.