Equivalences of (co)module algebra structures over Hopf algebras

We introduce the notion of support equivalence for (co)module algebras (over Hopf algebras), which generalizes in a natural way (weak) gradings. show that each class algebra structures on given A, there exists unique universal H together with an H-(co)module structure A such any other equivalent factors through action H. study and mentioned above group gradings, Hopf-Galois extensions, actions algebraic groups cocommutative algebras. how can be used to reduce classification problem (co)actions. apply asymptotic behaviour codimensions H-identities and, particular, analogue (formulated by Yu. A. Bahturin) Amitsur's conjecture, was originally concerned ordinary polynomial identities. As example we prove this all unital H-module $F[x]/(x^2)$ dual numbers.