Automorphisms and derivations of affine commutative and PI-algebras

We prove analogs of A. Selberg’s result for finitely generated subgroups of Aut ⁡ ( A ) \operatorname {Aut}(A) and of Engel’s theorem for subalgebras of Der ⁡ ( A ) \operatorname {Der}(A) for a finitely generated associative commutative algebra A A over an associative commutative ring. We prove also an analog of the theorem of W. Burnside and I. Schur about local finiteness of torsion subgroups of Aut ⁡ ( A ) \operatorname {Aut}(A) .