Skew derivations whose invariants satisfy a polynomial identity

If a is an automorphism and delta is a q-skew sigma-derivation of a ring R, then the subring of invariants is the set R-(delta) = {r is an element of R / delta(r) = 0}. The main result of this paper is THEOREM. Let R be a prime algebra with a q-skew sigma-derivation delta, whe re delta and sigma are algebraic. If R-(delta) satisfies a P.I., then R sat isfies a P.I. If delta is separable, then we also obtain the following result: THEOREM. Let delta be a separable q-skeut sigma-deviation of an algebra R, where delta and sigma are algebraic. (i) If R-(delta) satisfies a P.I., then R satisfies a P.I. (ii) If R-(sigma) boolean AND R-(delta) satisfies a P.I, and a is separable , then R satisfies a P.I. When R is a domain, it is necessary to assume neither that a is algebraic n or that delta is q-skew as we prove THEOREM. If R is a domain with an algebraic sigma-derivation delta such tha t R-(delta) satisfies a P.I., then R also satisfies a P.I. (C) 2000 Academi c Press.