is the direct limit of the GL("n"), which embeds in GL("n" + 1) as the upper left block matrix, and formula_16 is its commutator subgroup. Define an "elementary matrix" to be one which is the sum of an identity matrix and a single off-diagonal element (this is a subset of the elementary matrices used in linear algebra). Then Whitehead's lemma states that the group "E"("A") generated by elementary matrices equals the commutator subgroup GL("A"). Indeed, the group GL("A")/E("A") was first defined and studied by Whitehead, and is called the Whitehead group of the ring "A".