Automorphisms and derivations of affine commutative and PI-algebras

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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) .