Elementary matricies (Ex)
From Manifold Atlas
Let be an associative ring with unit and recall that an elementary matrix over is a square matrix of the form
where is the identity matrix, and is the matrix with zeros in all places except where it is and we have . Clearly each elementary matrix is invertible and so defines an element where
is the limit of the invertible matricies.
Exercise 0.1. Prove that , where is the subgroup generated by all elements in which are represented by elementary matrices.
Hint 0.2. For and write the matrix
as a product of elementary matrices
where is an matrix.
Recall that is defined to be the abelian group
Exercise 0.3. Prove that .