PHP decompositions and primitivity of group rings for linear groups

We show that every countable group $G$ that has Ozawa’s property $\mathrm{P}\mathrm{H}\mathrm{P}$ and contains a non-abelian free subgroup has the following property: the group ring $KG$ is primitive for any field $K$. As a consequence, we deduce that all non-trivial countable linear groups with trivial amenable radical have this property. Along the way, we prove that the class of groups with $\mathrm{P}\mathrm{H}\mathrm{P}$ enjoys some permanence properties that are of independent interest.