## 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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### References:

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