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# Steinberg group

Given an associative ring $R$ with identity, the Steinberg group $St(R)$ describes the minimal amount of relations between elementary matrices in $R$.

For $n\geq 3$, define $St_{n}(R)$ to be the free abelian group on symbols $x_{{ij}}(r)$ for $i,j$ distinct integers between $1$ and $n$, and $r\in R$, subject to the following relations:

$x_{{ij}}(r)x_{{ij}}(s)=x_{{ij}}(r+s)$ |

$[x_{{ij}},x_{{kl}}]=\begin{cases}1&\text{if $j\neq k$ and $i\neq l$}\\ x_{{il}}(rs)&\text{if $j=k$ and $i\neq l$}\\ x_{{kj}}(-sr)&\text{if $j\neq k$ and $i=l.$}\end{cases}$ |

Note that if $e_{{ij}}(r)$ denotes the elementary matrix with one along the diagonal, and $r$ in the $(i,j)$ entry, then the $e_{{ij}}(r)$ also satisfy the above relations, giving a well defined morphism $St_{n}(R)\to E_{n}(R)$, where the latter is the group of elementary matrices.

Taking a colimit over $n$ gives the Steinberg group $St(R)$. The importance of the Steinberg group is that the kernel of the map $St(R)\to E(R)$ is the second algebraic $K$-group of the ring $R$, $K_{2}(R)$. This also coincides with the kernel of the Steinberg group. One can also show that the Steinberg group is the universal central extension of the group $E(R)$.

## Mathematics Subject Classification

19C09*no label found*

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