THE GENERAL LINEAR GROUP OVER A RING

Han, Jun-Cheol

doi:10.4134/BKMS.2006.43.3.619

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THE GENERAL LINEAR GROUP OVER A RING

Journal title : Bulletin of the Korean Mathematical Society
Volume 43, Issue 3, 2006, pp.619-626
Publisher : The Korean Mathematical Society
DOI : 10.4134/BKMS.2006.43.3.619

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, ${$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.

Keywords

coprime ideals;general linear group of degree m over a ring;congruence relation ${${\equiv}_m(R)$}$ ;order of group;

Language

English

Cited by

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References

T. W. Hungerford, Algebra, Springer-Verlag, New York-Belin, 1980

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