Let $R$ be a unital ring with multiplicative identity $1_R$. For each $n \ge 1$, let $I_n \in GL_n(R)$ be the identity matrix, and for $i \ne j$ and $r \in R$ let $e_{ij}(r) \in GL_n(R)$ denote the elementary matrix obtained from $I_n$ by putting $r$ in the $(i,j)$-entry. Let
under the standard stabilisation maps $A\mapsto \operatorname{diag}(A,1_R)$, and let $E(R)\leq GL(R)$ be the stable elementary subgroup generated by all such elementary matrices $e_{ij}(r)$. Then $E(R)$ is perfect:
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\begin{align*}
E(R)=[E(R),E(R)].
\end{align*}
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Moreover $E(R)$ is normal in $GL(R)$, so the [quotient group](/theorems/790) $GL(R)/E(R)$ is defined, and this quotient is abelian.