THE GENERAL LINEAR GROUP OVER A RING

THE GENERAL LINEAR GROUP OVER A RING

Let m be any positive integer, R be a ring with identity, $M_m(R)$ be the matrix ring of all m by m matrices eve. R and $G_m(R)$ be the multiplicative group of all n by n nonsingular matrices in $M_m(R)$. In this pape., the following are investigated: (1) for any pairwise coprime ideals ${I_1,\;I_2,\;...,\;I_n}$ in a ring R, $M_m(R/(I_1{\cap}I_2{\cap}...{\cap}I_n))$ is isomorphic to $M_m(R/I_1){\times}M_m(R/I_2){\times}...{\times}M_m(R/I_n);$ and $G_m(R/I_1){\cap}I_2{\cap}...{\cap}I_n))$ is isomorphic to $G_m(R/I_1){\times}G_m(R/I_2){\times}...{\times}G_m(R/I_n);$ (2) In particular, if R is a finite ring with identity, then the order of $G_m(R)$ can be computed.