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Multiple commutators of elementary subgroups: end of the line. (English) Zbl 1451.19004

Let \(A\), \(B\), \(C\), \(D\) be two-sided ideals in the associative ring with unit \(R\). Denote the ideal \( AB + BA \) by \(A \circ B \). Recall that \(E(n,A)\) is the subgroup of \(\mathrm{GL}(n,R)\) generated by elementary matrices \(e_{ij}(x)\) with \(x \in A\). The key results in the paper are the following two lemmas, previously known for a class of rings containing all commutative rings.
Lemma 7. Let \(n \geq 3\). Then \( [[E(n,A),E(n,B)],E(n,C)]=[E(n,A \circ B),E(n,C)].\)
Lemma 8. Let \(n \geq 4\). Then \( [[E(n,A),E(n,B)],[E(n,C),E(n,D)]]=[E(n,A \circ B),E(n,C \circ D)].\)
There is also
Theorem 2. If \(A+B=R\) then \( [E(n,A),E(n,B)]=E(n,A \circ B).\)

MSC:

19B99 Whitehead groups and \(K_1\)
20H25 Other matrix groups over rings
20G35 Linear algebraic groups over adèles and other rings and schemes
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