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 ${\mathrm{S}}_{\mathrm{t}}(R)$, is defined by the following presentation:

• For every element $\lambda \in R$ and for $1\le i,j\le n$, $i\ne j$, 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 $i\ne k$.
  • $[e_{ij}(\lambda ),e_{kl}(\mu )]=1$ (i.e., is the identity element) for $i\ne l,j\ne k$.

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, this map is an isomorphism. Further, when $R$ is a field, it is also true that $E_n(R)$ coincides with the special linear group $S{L}_n(R)$ (see Elementary matrices of the first kind generate the special linear group over a field). In particular, the presentation described above gives a presentation for the special linear group over a field.

Note that $E_n(R)$ coinciding with $S{L}_n(R)$ also holds when $R$ is a Euclidean domain.