The Lasker–Noether theorem for commutative and noetherian module algebras over a pointed Hopf algebra

Let H be a pointed Hopf algebra over a field, let A be a commutative noetherian H-module algebra, and let I be an invariant ideal in A such that g(P) P for any group-like element g H and any associated prime P Ass(I). We prove that I admits an irredundant primary decomposition I=Q1∩ ∩Qn such that each Qi is invariant. Moreover, we introduce the concept of a convolutionally Hopf algebra and show that each associated prime of the ideal I is invariant, provided the Hopf algebra H is convolutionally reduced. Also it will be proved that in characteristic 0 every connected Hopf algebra is convolutionally reduced.