Lower K-groups

Definition

Let $R$ be a ring with identity. Let $GL(R) = \bigcup_{n \ge 1} GL_n(R)$, where $GL_n(R)$ is the general linear group and $g \mapsto \operatorname{diag}(g, 1)$ embeds $GL_n(R)$ in $GL_{n+1}(R)$. Let $E(R) \subseteq GL(R)$ be the subgroup generated by the elementary matrices $e_{ij}(r)$, $i \ne j$, $r \in R$. The matrix $e_{ij}(r)$ has $1$ on the diagonal, $r$ in position $(i, j)$ and $0$ elsewhere. The lower K-groups of $R$ are the following three abelian groups.

1. $K_0(R)$, the Grothendieck group of finitely generated projective left $R$-modules under direct sum.
2. $K_1(R)$ $= GL(R)/E(R)$.
3. $K_2(R)$ $= \ker(\mathrm{St}(R) \to E(R))$. Here the Steinberg group $\mathrm{St}(R)$ has generators $x_{ij}(r)$, $i \ne j$ positive integers, $r \in R$, and the following relations. They are $x_{ij}(r) x_{ij}(s) = x_{ij}(r + s)$, $[x_{ij}(r), x_{jk}(s)] = x_{ik}(rs)$ for $i \ne k$, and $[x_{ij}(r), x_{kl}(s)] = 1$ for $j \ne k$ and $i \ne l$. The map sends $x_{ij}(r)$ to $e_{ij}(r)$.