Group generated by elementary matrices over a unital ring

Definition

Let $R$ be a unital ring and $n$ be a natural number. The group generated by elementary matrices of degree $n$ over $R$, denoted $E_{n}(R)$, is defined as the subgroup of the general linear group $GL_{n}(R)$ generated by the elementary matrices $e_{ij}(\lambda )$ for $1\leq i,j\leq n$, $i\neq j$, and $\lambda \in R$. The elementary matrix $e_{ij}(\lambda )$ is a matrix with $1$s on the diagonal, $\lambda$ in the $(ij)^{th}$ position, and $0$s elsewhere.

There is a natural homomorphism from the Steinberg group over a unital ring $St_{n}(R)$ to this group. For a commutative unital ring and also for a division ring, we can define a determinant homomorphism and a special linear group $SL_{n}(R)$, and $E_{n}(R)$ is a subgroup of the special linear group. For a field, division ring, or Euclidean domain, $E_{n}(R)$ equals $SL_{n}(R)$.