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COMMUTATORS OF ELEMENTARY SUBGROUPS: CURIOUSER AND CURIOUSER

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Abstract

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 zij (ab, c) = tij (c)tji(ab)tij (–c) and zij (ba, c), and

(2) the elementary commutators [tij (a), tji(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 [tij (a), thk(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.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, St. Petersburg State University, St. Petersburg, Russia

    N. VAVILOV

  2. Department of Mathematics, Beijing Institute of Technology, Beijing, China

    Z. ZHANG

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  1. N. VAVILOV
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  2. Z. ZHANG
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N. Vavilov is the work of the first author was supported by the Russian Science Foundation grant 17-11-01261.

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VAVILOV, N., ZHANG, Z. COMMUTATORS OF ELEMENTARY SUBGROUPS: CURIOUSER AND CURIOUSER. Transformation Groups 28, 487–504 (2023). https://doi.org/10.1007/s00031-021-09662-z

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