Invariants and rings of quotients of $H$-semiprime $H$-module algebra satisfying a polynomial identity

We consider an action of a finite-dimensional Hopf algebra $H$ on a PI-algebra. We prove that an $H$-semiprime $H$-module algebra $A$ has a Frobenius artinian classical ring of quotients $Q$ if $A$ has a finite set of $H$-prime ideals with zero intersection. The ring of quotients $Q$ is an $H$-semisimple $H$-module algebra and finitely generated module over the subalgebra of central invariants. Moreover, if the algebra $A$ is projective module of constant rank over its center then $A$ is integral over the subalgebra of central invariants.