On the normal structure of the general linear group over a ring

ON THE NORMAL STRUCTURE OF THE

GENERAL LINEAR GROUP OVER A RING

ALEXEI STEPANOV

Chair of Mathematics II

St.Petersburg Electrotechnical University

194356, Prof. Popov st. 5, St.Petersburg, Russia

Introduction

The study of the normal structure of the general linear group over general rings was

initiated by H.Bass [Bs1], [Bs2] [Bs3] in mid 60-th. In particular he established a complete

description of the normal subgroups of the quite general linear group over an arbitrary

ring. Also H.Bass introduced a new notion of dimension of rings, the so called stable rank,

and discovered that the principal structure results remain valid for groups whose degree

is large with respect to the stable rank. J.Milnor initiated the study of the Steinberg

group in this context [M]. Very important contributions in this direction are due to A.Bak,

R.K.Dennis, W.van der Kallen, A.A.Suslin and L.N.Vaserstein.

The next major breakthrough was triggered by the works of J.S.Wilson [W], I.Z.Golub￾chik [Gl] and A.A.Suslin [S] who discovered that for commutative rings the above results

remain valid starting with degree 3 independently of the dimension of the ground ring.

Further results in this direction were obtained by Z.I.Borewich, I.Z.Golubchik, V.I.Kopei￾ko, A.V.Mikhalev, A.A.Suslin, G.Taddei, L.N.Vaserstein, N.A.Vavilov and many others.

W. van der Kallen [vdK] and M.S.Tulenbaev [T] proved the analogous results for the

Steinberg group (starting with dimension 4). Also it should be mentioned that V.N.Ge￾rasimov constructed (in any dimension) an example of a ring such that the elementary

subgroup is not normal in the general linear group (and therefore, the normal structure of

the general linear group over such a ring is nonstandard). A complete review of this subject

is given in the book by A.J.Hahn and O.T.O’Meara [HO’M] and in survey by N.A.Vavilov

[Vv].

The most intriguing problem in the theory of linear groups over rings nowadays is to

determine the exact class of rings where the behaviour of the general linear group and the

The author gratefully acknowledges support of St.Petersburg Mayor’s Office for the grant for young

scientists and the SFB 343 an der Universit¨at Bielefeld.

The reseach described in this publication was made possible in part by Grant N. JHP100 from the

International Science Foundation and Russian Goverment.

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