Steinberg group over a unital ring

Definition

Suppose $R$ is a (associative) unital ring and $n$ is a natural number. The Steinberg group of degree $n$ over $R$, denoted $\operatorname {St} _{n}(R)$, is defined by the following presentation:

• For every element $\lambda \in R$ and for $1\leq i,j\leq n$, $i\neq j$, we have a generator $e_{ij}(\lambda )$.
• The relations are as follows:
  • $e_{ij}(\lambda )e_{ij}(\mu )=e_{ij}(\lambda +\mu )$ (note that this in particular implies that $e_{ij}(0)$ is the identity element for all $i,j$.
  • $[e_{ij}(\lambda ),e_{jk}(\mu )]=e_{ik}(\lambda \mu )$ for $i\neq k$.
  • $[e_{ij}(\lambda ),e_{kl}(\mu )]=1$ (i.e., is the identity element) for $i\neq l,j\neq k$.

The stable Steinberg group for a unital ring is similar to the above except that we have no size restrictions on $i$ and $j$.

Facts

For every $R$ and $n$, there is a standard homomorphism from the Steinberg group to the group generated by elementary matrices over a unital ring $E_{n}(R)$. This homomorphism sends the generator $e_{ij}(\lambda )$ to the elementary matrix $e_{ij}(\lambda )$, i.e., the matrix with $1$s on the diagonal, $\lambda$ in the $(ij)^{th}$ entry, and $0$s elsewhere. When $R$ is a field, the group $E_{n}(R)$ coincides with the special linear group $SL_{n}(R)$ (see Elementary matrices of the first kind generate the special linear group over a field).

Note that $E_{n}(R)$ coinciding with $SL_{n}(R)$ also holds when $R$ is a Euclidean domain.