Algebraic K-theory is the branch of algebra dealing with linear algebra （especially in the limiting case of large matrices） over a general ring R instead of over a field. It associates to any ring R a sequence of abelian groups Ki（R）. The first two of these， K0 and K1， are easy to describe in concrete terms， the others are rather mysterious. For instance， a finitely generated projective R-module defines an element of K0（R）， and an invertible matrix over R has a "determinant" in K1（R）. The entire sequence of groups K1（R） behaves something like a homology theory for rings.
　　Algebraic K-theory plays an important role in many areas， especially number theory， algebraic topology， and algebraic geometry. For instance， the class group of a number field is essentially K0（R）， where R is the ring of integers， and "Whitehead torsion" in topology is essentially an element of K1（Zπ）， where π is the fundamental group of the space being studied. K-theory in algebraic geometry is basic to Grothendieck's approach to the Riemann-Roch problem. Some formulas in operator theory， involving determinants and determinant pairings， are best understood in terms of algebraic K-theory. There is also substantial evidence that the higher K-groups of fields and of rings of integers are related to special values of L-functions and encode deep arithmetic information.