  An algebra A with a generalized H-action is a generalization of an H-module
algebra where H is just an associative algebra with 1 and a relaxed
compatibility condition between the multiplication in A and the H-action on A
holds. At first glance this notion may appear too general, however it enables
to work with algebras endowed with various kinds of additional structures (e.g.
(co)module algebras over Hopf algebras, graded algebras, algebras with an
action of a (semi)group by (anti)endomorphisms). This approach proves to be
especially fruitful in the theory of polynomial identities. We show that if A
is a finite dimensional (not necessarily associative) algebra over a field of
characteristic 0 and A is simple with respect to a generalized H-action, then
there exists an exponent of the codimension growth of polynomial H-identities
of A. In particular, if A is a finite dimensional (not necessarily group
graded) graded-simple algebra, then there exists an exponent of the codimension
growth of graded polynomial identities of A. In addition, we study the
free-forgetful adjunctions corresponding to (not necessarily group) gradings
and generalized H-actions.
