  Necessary and sufficient conditions for an Ore extension $S=R[x;\si,\de]$ to
be a {\rm PI} ring are given in the case $\si$ 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.
