Actions of pointed Hopf algebras with reduced pi invariants

Let $R$ be an $H$-module algebra, where $H$ is a pointed Hopf algebra acting on $R$ finitely of dimension $N$. Suppose that $L^H\neq 0$ for every nonzero $H$-stable left ideal of $R$. It is proved that if $R^H$ satisfies a polynomial identity of degree $d$, then $R$ satisfies a polynomial identity of degree $dN$ provided at least one of the following additional conditions is fulfilled:

1. $R$ is semiprime and $R^H$ is almost central in $R$,
2. $R$ is reduced.

If we also assume that $R^H$ is central, then $R$ satisfies the standard polynomial identity of degree $2[\sqrt{N}]$, where $[\sqrt{N}]$ is the greatest integer in $\sqrt{N}$.