On \(K_2\)-group of a formal matrix ring (Q1943215)

The authors give a presentation of Milnor's \(K_2\) group of a formal matrix ring, under the assumption that the off-diagonal entries form an ideal. More precisely, assume (for \(i,j=1,\dots,t\)) that \(S_{ii}\) are rings and \(S_{ij}\) are \(S_{ii},S_{jj}\) bimodules endowed with composition maps \(S_{ij} \otimes S_{jk} \rightarrow S_{ik}\) that make \(S = \sum S_{ij}\) an associative ring. The typical example would be a semiperfect ring. Assume further that \(S_{ij}S_{ji} = 0\), so that \(I = \sum_{i\neq j} S_{ij}\) is a split radical ideal. Then the short exact sequence \[ 0 \longrightarrow K_2(R,I) \longrightarrow K_2(R) \longrightarrow K_2(R/I) \longrightarrow 0 \] splits. In what follows the notation \(x_{\alpha\beta}\) will always refer to an element of \(S_{\alpha\beta}\). The authors show that \(K_2(R,I)\) is generated by the symbols \(\langle a_{ij},b_{ji} \rangle\) (\(i\neq j\)), subject to the following sets of relations: antisymmetry, biadditivity, transfer of scalars (i.e., \(\langle a_{ij}r_{jj},b_{ji} \rangle = \langle a_{ij},r_{jj}b_{ji} \rangle\)) and \(\langle a_{ij},b_{jk}c_{ki} \rangle \langle b_{jk},c_{ki}a_{ij} \rangle \langle c_{ki},a_{ij}b_{jk}\rangle=1\) for distinct \(i,j,k\).