We prove that if G(R) = Gπ(Φ , R) (E(R) = Eπ(Φ , R)) is an (elementary) Chevalley group of rank > 1 , R is a local ring (with 12 for the root systems A2, Bl, Cl, F4, G2 and with 13 for G2) , then the group G(R) (or (E(R)) is regularly bi-interpretable with the ring R. As a consequence of this theorem, we show that the class of all Chevalley groups over local rings (with the listed restrictions) is elementarily definable, i.e., if for an arbitrary group H we have H≡ Gπ(Φ , R) , then there exists a ring R′≡ R such that H≅ Gπ(Φ , R′) .
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- Chevalley groups
- Elementary definability
- Local rings
- Regular bi-interpretability