Groupoid gradings: A Morita context, semiprimality, and a Bergman-type question

In this paper, we are interested in some natural consequences arising from the canonical duality relation between the concepts of groupoid grading and groupoid action. Any algebra A over a field K graded by a finite groupoid has a canonical structure of a module algebra over the dual algebra of the groupoid algebra . The results here presented concern to the multiplicative structure of the smash product algebra and its representations. An application is the affirmative answer to a Bergmann-type question for groupoid gradings.