# Group generated by elementary matrices over a unital ring

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 $1 \le i,j \le n$, $i \ne j$, 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 $SL_n(R)$.