It is proved that if { M n } \left \{ {{M_n}} \right \} is a stable system of coefficients for G l n ( R ) {\text {G}}{{\text {l}}_n}\left ( R \right ) and H 0 ( Gl ( R ) , lim ( M n ) ) {H_0}\left ( {{\text {Gl}}\left ( R \right ),{\text {lim}}\left ( {{M_n}} \right )} \right ) contains Z {\mathbf {Z}} , then for any j j , the group H j ( Gl ( R ) , lim ( M n ) ) {H_j}\left ( {{\text {Gl}}\left ( R \right ),{\text {lim}}\left ( {{M_n}} \right )} \right ) contains H j ( Gl ( R ) , Z ) {H_j}\left ( {{\text {Gl}}\left ( R \right ),Z} \right ) as a direct summand. Now let Gl ( Z ) {\text {Gl}}\left ( {\mathbf {Z}} \right ) act on M ( Z ) M\left ( {\mathbf {Z}} \right ) (matrices over Z {\mathbf {Z}} ) by conjugation. Then our theorem implies that the trace map tr: M ( Z ) → Z {\text {tr:}}M\left ( {\mathbf {Z}} \right ) \to {\mathbf {Z}} is a split epimorphism on homology.