Real strict localizations
After Cohn a commutative ring R is called a GE\({}_ n\)-ring if and only if \(SL_ n(R)\) is generated by elementary matrices, where \(n\geq 2\). Suslin and Cohn have shown that the polynomial ring \({\mathbb{Z}}[t_ 1,...,t_ d]\) is \(GE_ n\) if and only if \(n\geq 3\), where \({\mathbb{Z}}\) is the ring of rational integers. In this paper the authors consider the following question: Given algebraic numbers \(\alpha_ 1,...,\alpha_ d\), for which n (if any) is the ring \(A={\mathbb{Z}}[\alpha_ 1,...,\alpha_ d]\) a \(GE_ n\)-ring? By standard results from algebraic K-theory it follows that (a) A is \(GE_ n\), for all \(n\geq 2\), (b) A is not \(GE_ n\), for any \(n\geq 2\) or (c) A is \(GE_ n\) if and only if \(n\geq 3\). Using results of Bass, Milnor, Serre and Liehl the authors prove that the index \(| SL_ n(A):E_ n(A)|\) divides m, provided that \(n\geq 3\) or that A has infinitely many units, where m is the number of roots of unity in the smallest number field containing A and \(E_ n(A)\) is the subgroup of \(SL_ n(A)\) generated by elementary matrices. They prove further that A is of type (a) when, for example, at least one \(\alpha_ i\) is not an algebraic integer, each \(\alpha_ i\) is real or each \(\alpha_ i\) is a root of unity. There are examples of type (b) which are due to Liehl. In particular when \(A={\mathbb{Z}}+(m^ 2)\), where \((m^ 2)\) is the principal ideal generated by \(m^ 2\) in the ring of integers of a totally imaginary number field (which has m roots of unity), then \(| SL_ n(A):E_ n(A)| =m\), when \(n\geq 3\) or A has infinitely many units. It is known that a ring of integers of an imaginary quadratic number field which is not euclidean is of type (c). (Only five such rings are euclidean.) The authors provide non-integrally closed examples of type (c).
scientific article; zbMATH DE number 3957453
scientific article