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Finitely generated GEn-rings

Published online by Cambridge University Press:  14 November 2011

A. W. Mason
A. W. Mason
Affiliation:
Department of Mathematics, The University, Glasgow G12 8QW, U.K.
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Synopsis

Suslin and Cohn have proved that the polynomial ring Z[t1, …,td], where Z is the ring of rational integers and d >0, is a GEn-ring if and only if n ≧ 3. (A commutative ring R with identity is called a GEn-ring if and only if SLn(R) is generated by elementary matrices, where n ≧ 2.) In this paper we consider the following question:

Given algebraic numbers α1, …,αd, for which n (if any) is the ring A =Z [α1, …,αd a GEn-ring?

By standard results from algebraic K-theory it follows that (a) A is GEn for all n ≧ 2, (b) A is not GEn for any n ≧ 2, or (c) A is GEn if and only if n ≧ 3. Examples of each type are provided. In particular, it is shown that if each αi is real or at least one αi is not an algebraic integer, then A is of type (a).

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 106 , Issue 1-2 , 1987 , pp. 63 - 71
Copyright
Copyright © Royal Society of Edinburgh 1987

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