Defining \(R\) and \(G(R)\). (English) Zbl 1521.20111
Summary: We show that for Chevalley groups \(G ( R )\) of rank at least 2 over an integral domain \(R\) each root subgroup is (essentially) the double centralizer of a corresponding root element. In many cases, this implies that \(R\) and \(G ( R )\) are bi-interpretable, yielding a new approach to bi-interpretability for algebraic groups over a wide range of rings and fields.
For such groups it then follows that the group \(G ( R )\) is (finitely) axiomatizable in the appropriate class of groups provided \(R\) is (finitely) axiomatizable in the corresponding class of rings.
For such groups it then follows that the group \(G ( R )\) is (finitely) axiomatizable in the appropriate class of groups provided \(R\) is (finitely) axiomatizable in the corresponding class of rings.
MSC:
| 20G35 | Linear algebraic groups over adèles and other rings and schemes |
| 03C60 | Model-theoretic algebra |
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