Ore extensions satisfying a polynomial identity

Necessary and sufficient conditions for an Ore extension $S=R[x;\text{si},\text{de}]$ to be a {\rm PI} ring are given in the case $\text{si}$ is an injective endomorphism of a semiprime ring $R$ satisfying the {\rm ACC} on annihilators. Also, for an arbitrary endomorphism $\tau$ of $R$, a characterization of Ore extensions $R[x;\tau ]$ which are {\rm PI} rings is given, provided the coefficient ring $R$ is noetherian.