Let $R$ be a unital ring. For each integer $n \ge 1$, let $M_n(R)$ denote the ring of $n \times n$ matrices over $R$, let $I_n \in M_n(R)$ denote the identity matrix, and let $GL_n(R)$ denote the group of invertible elements of $M_n(R)$. For $i \ne j$ and $r \in R$, let $e_{ij}^{(n)}(r) \in GL_n(R)$ denote the elementary matrix whose diagonal entries are $1_R$, whose $(i,j)$-entry is $r$, and whose remaining off-diagonal entries are $0_R$. When the ambient matrix size $n$ is clear, write $e_{ij}(r)$ for $e_{ij}^{(n)}(r)$. Let

\begin{align*} GL(R)=\varinjlim_{n} GL_n(R) \end{align*}

with transition maps $A \mapsto \operatorname{diag}(A,1_R)$, and let $E_n(R) \le GL_n(R)$ be the subgroup generated by the matrices $e_{ij}^{(n)}(r)$ with $i \ne j$ and $r \in R$. Let $E(R) \le GL(R)$ be the subgroup generated by the images of all $E_n(R)$ under the canonical maps $GL_n(R) \to GL(R)$. Then $E(R)$ is a [normal subgroup](/page/Normal%20Subgroup) of $GL(R)$.