Group generated by elementary matrices over a unital ring
Let be a unital ring and be a natural number. The group generated by elementary matrices of degree over , denoted , is defined as the subgroup of the general linear group generated by the elementary matrices for , , and . The elementary matrix is a matrix with s on the diagonal, in the position, and s elsewhere.
There is a natural homomorphism from the Steinberg group over a unital ring to this group. For a commutative unital ring and also for a division ring, we can define a determinant homomorphism and a special linear group , and is a subgroup of the special linear group. For a field, division ring, or Euclidean domain, equals .