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

Let R be a commutative domain of stable range 1 with 2 a unit. In this paper we describe the homomorphisms between SL2(R) and GL2(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 GL2(R) and GL2(K). Those homomorphisms are found of either extensions of homomorphisms from SL2(R) to GL2(K) or the products of inner automorphisms with certain group homomorphisms from GL2(R) to K.