Prestabilization for K₁ of banach algebras

The injective stability for the general linear group modulo elementary matrices begins at one plus the stable range of the ring of entries. At one step earlier, the kernel of stabilization is perhaps larger than the group of elementary matrices. Using a Dieudonné-style determinant, it is shown that this kernel is generated by matrices of the form (1 + XY) (1 + XY), under certain conditions on the ring of entries, or in the relative case, on the ideal. For any ideal of stable rank one, the kernel is given in terms of generators (X + Z + XYZ) (X + Z + ZYX). Under somewhat stronger conditions, the kernel is shown to be a commutator subgroup.