On the semiprimitivity and the semiprimality problems for partial smash products

In this paper we discuss about the semiprimitivity and the semiprimality of partial smash products. Let $H$ be a semisimple Hopf algebra over a field $\mathbb{k}$ and let $A$ be a left partial $H$-module algebra. We study the $H$-prime and the $H$-Jacobson radicals of $A$ and its relations with the prime and the Jacobson radicals of $\underline{A\#H}$, respectively. In particular, we prove that if $A$ is $H$-semiprimitive, then $\underline{A\#H}$ is semiprimitive provided that all irreducible representations of $A$ are finite-dimensional, or $A$ is an affine PI-algebra over $\mathbb{k}$ and $\mathbb{k}$ is a perfect field, or $A$ is locally finite. Moreover, we prove that $\underline{A\#H}$ is semiprime provided that $A$ is an $H$-semiprime PI-algebra, generalizing for the setting of partial actions, the main results of [20] and [19].