Abstract
Let R be a commutative ring with 1, A, B ⊴ R be its ideals, GL(n, R, A) be the principal congruence subgroup of level A in GL(n, A), and E(n, R, A) be the relative elementary subgroup of level A. We prove the following commutator formula $$ [E(n,R,A),GL(n,R,B)] = [E(n,R,A),E(n,R,B)], $$ which generalizes known results. The proof is yet another variation on the theme of decomposition of unipotents.
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