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Standard commutator formula, revisited. (English. Russian original) Zbl 1254.20038

Vestn. St. Petersbg. Univ., Math. 43, No. 1, 12-17 (2010); translation from Vestn. St-Peterbg. Univ., Ser. I, Mat. Mekh. Astron. 2010, No. 1, 16-22 (2010).
Summary: Let \(R\) be an associative ring with \(1\); \(A,B\trianglelefteq R\) be its ideals; \(C(n,R,A)\) be the full congruence subgroup of level \(A\) in \(\mathrm{GL}(n,R)\); and \(E(n,R,A)\) be the relative elementary subgroup of level \(A\). We present a very easy proof of the following commutator formula: \[ [E(n,R,A),C(n,R,B)]=[E(n,R,A),E(n,R,B)] \] for all commutative rings based exclusively on the absolute standard commutator formula and its non-commutative counterparts. This generalizes and strengthens Mason and Stothers’ results, our results, and those of Hazrat and Zhang. For comaximal ideals \(A+B=R\), we show that this commutator is \(E(n,R,AB+BA)\).

MSC:

20G35 Linear algebraic groups over adèles and other rings and schemes
20F12 Commutator calculus
20H25 Other matrix groups over rings
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