Splitting theorem for homology of ${\rm GL}(R)$

It is proved that if $\left\{ {{M_n}} \right\}$ is a stable system of coefficients for ${\text{G}}{{\text{l}}_n}\left( R \right)$ and ${H_0}\left( {{\text{Gl}}\left( R \right),{\text{lim}}\left( {{M_n}} \right)} \right)$ contains ${\mathbf{Z}}$, then for any $j$, the group ${H_j}\left( {{\text{Gl}}\left( R \right),{\text{lim}}\left( {{M_n}} \right)} \right)$ contains ${H_j}\left( {{\text{Gl}}\left( R \right),Z} \right)$ as a direct summand. Now let ${\text{Gl}}\left( {\mathbf{Z}} \right)$ act on $M\left( {\mathbf{Z}} \right)$ (matrices over ${\mathbf{Z}}$ ) by conjugation. Then our theorem implies that the trace map ${\text{tr:}}M\left( {\mathbf{Z}} \right) \to {\mathbf{Z}}$ is a split epimorphism on homology.