Commutators of Relative and Unrelative Elementary Groups, Revisited
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COMMUTATORS OF RELATIVE AND UNRELATIVE ELEMENTARY GROUPS, REVISITED N. Vavilov∗ and Z. Zhang†
UDC 512.5
Let R be any associative ring with 1, let n ≥ 3, and let A, B be two-sided ideals of R. In the present paper, we show that the mixed commutator subgroup [E(n, R, A), E(n, R, B)] is generated as a group by the elements of the two following forms: 1) zij (ab, c) and zij (ba, c), 2) [tij (a), tji (b)], where 1 ≤ i = j ≤ n, a ∈ A, b ∈ B, c ∈ R. Moreover, for the second type of generators, it suffices to fix one pair of indices (i, j). This result is both stronger and more general than the previous results by Roozbeh Hazrat and the authors. In particular, it implies that for all associative rings one has the equality [E(n, R, A), E(n, R, B)] = [E(n, A), E(n, B)], and many further corollaries can be derived for rings subject to commutativity conditions. Bibliography: 36 titles.
To the remarkable St.Petersburg algebraist Alexander Generalov 1. Introduction In the present note, we generalize and strengthen the results by Roozbeh Hazrat and the authors [13, 15, 28] on generation of mutual commutator subgroups of relative and unrelative elementary subgroups in the general linear group. Namely, we both dramatically reduce the sets of generators that occur therein and either seriously weaken, or completely remove commutativity conditions. Let R be an associative ring with 1, and GL(n, R) be the general linear group of degree n ≥ 3 over R. As usual, e denotes the identity matrix, whereas eij denotes a standard matrix unit. For c ∈ R and 1 ≤ i = j ≤ n, we denote by tij (c) = e+ ceij the corresponding elementary transvection. To an ideal A R, we assign the elementary subgroup E(n, A) = tij (a), a ∈ A, 1 ≤ i = j ≤ n. The corresponding relative elementary subgroup E(n, R, A) is defined as the normal closure of E(n, A) in the absolute elementary subgroup E(n, R). From the work of Michael Stein, Jacques Tits, and Leonid Vaserstein it is classically known that as a group E(n, R, A) is generated by zij (a, c) = tji (c)tij (a)tji (−c), where 1 ≤ i = j ≤ n, a ∈ A, c ∈ R. Further, consider the reduction homomorphism ρI : GL(n, R) −→ GL(n, R/I) modulo I. By definition, the principal congruence subgroup GL(n, I) = GL(n, R, I) is the kernel of ρI . In other words, GL(n, I) consists of all matrices g congruent to e modulo I. A first version of the following result was discovered (in a slightly less precise form) by Roozbeh Hazrat and the second author, see [15, Lemma 12]. In exactly this form it is stated in our paper [13, Theorem 3A]. Theorem A. Let R be a quasi-finite ring with 1, let n ≥ 3, and let A, B be two-sided ideals of R. Then the mixed commutator subgroup [E(n, R, A), E(n, R, B)] is generated as a group by the elements of the form • zij (ab, c) and zij (ba, c), • [tij (a), tji (b)], ∗ †
St.Petersburg State University, St.Petersburg, Russia, e-mail: [email protected]. Beijing Institute of Technology, Beijing, China, e-mail: [email protected].
Published in Zapiski Nauchnykh Seminarov POMI, Vol. 485, 2019, pp. 58–71.
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