COMMUTATORS OF ELEMENTARY SUBGROUPS: CURIOUSER AND CURIOUSER

Let R be any associative ring with 1, n ≥ 3, and let A, B be two-sided ideals of R. In our previous joint works with Roozbeh Hazrat [17], [15], we have found a generating set for the mixed commutator subgroup [E(n, R, A); E(n, R, B)]. Later in [29], [34] we noticed that our previous results can be drastically improved and that [E(n, R, A); E(n, R, B)] is generated by (1) the elementary conjugates z_ij (ab, c) = t_ij (c)t_ji(ab)t_ij (–c) and z_ij (ba, c), and (2) the elementary commutators [t_ij (a), t_ji(b)], where 1 ≤ i ≠= j ≤ n, a ∈ A, b ∈ B, c ∈ R. Later in [33], [35] we noticed that for the second type of generators, it even suffices to fix one pair of indices (i, _j). Here we improve the above result in yet another completely unexpected direction and prove that [E(n, R, A); E(n, R, B)] is generated by the elementary commutators [t_ij (a), t_hk(b)] alone, where 1 ≤ i ≠ = j ≤ n, 1 ≤ h ≠ = k ≤ n, a ∈ A, b ∈ B. This allows us to revise the technology of relative localisation and, in particular, to give very short proofs for a number of recent results, such as the generation of partially relativised elementary groups E(n, A)E^{(n, B)}, multiple commutator formulas, commutator width, and the like.