On the generators of elementary subgroups of general linear groups

Let R be a ring with identity and let Eij ∈ Mn(R) be the usual n X n matrix units, where n ≥ 2 and 1≤i, j≤N. Let En(R) be the subgroup of GLn(R) generated by all Tij(q where r ∈ R and i ≄ j. For each (two-sided) R-ideal q let En(R, q) be the normal subgroup of En(R) generated by Tij(q), where q ∈ q. The subgroup En(R, q) plays an important role in the theory of GLn(R). For example, Vaserˇstein has proved that, for a larger class of rings (which includes all commutative rings), every subgroup S of GLn(R), when R ∈ and n≥3, contains the subgroup En(R, q0), where q0 is the R-ideal generated by αij, rαij-αjjr (i ≄ j, r ∈ R), for all (αij) ∈ S. (See [13, Theorem 1].) In addition Vaseršstein has shown that, for the same class of rings, En(R, q) has a simple set of generators when n ≥ 3. Let Ên(R, q) be the subgroup of En(R, q) generated by Tij(r)Tij(q)Tij(−r), where r ∈ R, q ∈ q. Then Ên(R, q) = En(R, q), for all q, when R ∈ and n ≥ 3.(See [13, Lemma 8].)