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 not equal= j <= n, a is an element of A, b is an element of B, c is an element of 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 not equal = j <= n, 1 <= h not equal = k <= n, a is an element of A, b is an element of 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,E- B), multiple commutator formulas, commutator width, and the like.