Homomorphisms of two-dimensional linear groups over a ring of stable range one

Abstract

Let R be a commutative domain of stable range 1 with 2 a unit. In this paper we describe the homomorphisms between SL 2 ( R ) and GL 2 ( K ) where K is an algebraically closed field. We show that every non-trivial homomorphism can be decomposed uniquely as a product of an inner automorphism and a homomorphism induced by a morphism between R and K. We also describe the homomorphisms between GL 2 ( R ) and GL 2 ( K ) . Those homomorphisms are found of either extensions of homomorphisms from SL 2 ( R ) to GL 2 ( K ) or the products of inner automorphisms with certain group homomorphisms from GL 2 ( R ) to K.