Presentations of general linear groups

Abstract

Let R be an associative ring with a 1 . Denote by GLn(R) the group of invertible nxn matrices over R, and by GEn(R) the subgroup of GLp(r) generated by the elementary and invertible diagonal matrices. Certain specified relations between these generators hold universally, that is, for any ring R. We call a ring R universal for GEn if GEn(R)'has these relations as defining relations, and we shew that if R is a local ring (i.e. a ring in which the set of all non-units is an ideal) or the ring of rational integers, then R is universal for GEn, for all n. This both generalizes known results for n=2, and includes the classical case where R is a field, possibly skew. By adding further relations to those already, considered we obtain in a similar way the concept 'quasi-universal for GE? ', giving a class of rings which strictly includes the class of all rings universal for GEp, but which is better behaved than the latter under certain ring constructions. We shew that every semi-local ring (i.e. every ring R such that R modulo its Jacobson radical has the minimum condition on right ideals) is quasi-universal for GEn , for all n. Finally we shew how to obtain a presentation of GEn(R) for any R. This is unwieldy, but simplifies greatly for a certain class of rings called GE2-reducible rings, which includes all Euclidean rings. We shew that for such rings R a set of defining relations for GEn(R), for n > 3, is obtained by taking the universal relations together with certain relations in GEa(R).