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On the generators of elementary subgroups of general linear groups

Published online by Cambridge University Press:  18 May 2009

A. W. Mason
Affiliation:
Department of Mathematics, University of Glasgow, Glasgow G12 8QW, Scotland e-mail: awm@MATHS.GLA.AC.UK.
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Let R be a ring with identity and let EijMn(R) be the usual n X n matrix units, where n ≥ 2 and 1≤i, jN. Let En(R) be the subgroup of GLn(R) generated by all Tij(q where rR and i ≄ j. For each (two-sided) R-ideal q let En(R, q) be the normal subgroup of En(R) generated by Tij(q), where qq. The subgroup En(R, q) plays an important role in the theory of GLn(R). For example, Vaserˇstein has proved that, for a larger class of rings (which includes all commutative rings), every subgroup S of GLn(R), when R and n≥3, contains the subgroup En(R, q0), where q0 is the R-ideal generated by αij, rαijjjr (i ≄ j, rR), for all (αij) ∈ S. (See [13, Theorem 1].) In addition Vaseršstein has shown that, for the same class of rings, En(R, q) has a simple set of generators when n ≥ 3. Let Ên(R, q) be the subgroup of En(R, q) generated by Tij(r)Tij(q)Tij(−r), where rR, qq. Then Ên(R, q) = En(R, q), for all q, when R and n ≥ 3.(See [13, Lemma 8].)

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 1996

References

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