Defining R and G(R).

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 biinterpretability 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. [ABSTRACT FROM AUTHOR]