THE GENERAL LINEAR GROUP OVER A RING

Title & Authors
THE GENERAL LINEAR GROUP OVER A RING
Han, Jun-Cheol;

Abstract
Let m be any positive integer, R be a ring with identity, $\small{M_m(R)}$ be the matrix ring of all m by m matrices eve. R and $\small{G_m(R)}$ be the multiplicative group of all n by n nonsingular matrices in $\small{M_m(R)}$. In this pape., the following are investigated: (1) for any pairwise coprime ideals $\small{{I_1,\;I_2,\;...,\;I_n}}$ in a ring R, $\small{M_m(R/(I_1{\cap}I_2{\cap}...{\cap}I_n))}$ is isomorphic to $\small{M_m(R/I_1){\times}M_m(R/I_2){\times}...{\times}M_m(R/I_n);}$ and $\small{G_m(R/I_1){\cap}I_2{\cap}...{\cap}I_n))}$ is isomorphic to $\small{G_m(R/I_1){\times}G_m(R/I_2){\times}...{\times}G_m(R/I_n);}$ (2) In particular, if R is a finite ring with identity, then the order of $\small{G_m(R)}$ can be computed.
Keywords
coprime ideals;general linear group of degree m over a ring;congruence relation $\small{{\equiv}_m(R)}$;order of group;
Language
English
Cited by
1.
On the Zero Divisor Graphs of the Ring of Lipschitz Integers Modulo n, Advances in Applied Clifford Algebras, 2017, 27, 2, 1191
2.
Free Cyclic Submodules in the Context of the Projective Line, Results in Mathematics, 2016, 70, 3-4, 567
3.
On the distribution of eigenspaces in classical groups over finite rings, Linear Algebra and its Applications, 2014, 443, 50
4.
Fundamental domains for congruence subgroups of SL2 in positive characteristic, Journal of Algebra, 2011, 325, 1, 431
References
1.
T. W. Hungerford, Algebra, Springer-Verlag, New York-Belin, 1980

2.
B. R. McDonald, Finite Rings with Identity, Marcel Dekker, Inc, New York, 1974