Lifting units in self-injective rings and an index theory for Rickart C*-algebras

In this paper we study the following question: If R is a right self-injective ring and I an ideal of R, when can the units of R/I be lifted to units of R? We answer this question in terms of K0(I). For a purely infinite regular right self-injective ring R we obtain an isomorphism between K1(R/I) and K0(I) which can be viewed as an analogue of the index map for Fredholm operators. By giving a purely algebraic description of the connecting map K1(A/I) → K0(I) in the case where A is a Rickart C*-algebra, we are able to extend the classical index theory to Rickart C*-algebras in a way which also includes Breuer’s theory for W*-algebras. © 1987 by Pacific Journal of Mathematics.