  Let $R$ be any associative ring with $1$, $n\ge 3$, 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 $1\le i\neq j\le n$, $a\in A$, $b\in B$, $c\in
R$. 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
$\big[E(n,R,A),E(n,R,B)\big]=\big[E(n,A),E(n,B)\big]$ and many further
corollaries can be derived for rings subject to commutativity conditions.

Rings and Algebras;Group Theory