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Let R be any associative ring with 1, 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) z_ij (ab,c) and z_ij (ba,c), 2) [t_ij (a), t_ji(b)], where 1<= i 6/= 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.
Translated title of the contributionЕще раз о взаимных коммутантах относительных и настоящих элементарных групп
Original languageEnglish
Pages (from-to)339-348
Number of pages10
JournalJournal of Mathematical Sciences (United States)
Volume251
Issue number3
Early online date30 Oct 2020
DOIs
StatePublished - 1 Dec 2020

    Scopus subject areas

  • Mathematics(all)

ID: 61526731