Ore extensions satisfying a polynomial identity

TL;DR

This paper establishes necessary and sufficient conditions for Ore extensions to satisfy polynomial identities, focusing on cases where the endomorphism is injective or arbitrary, with specific ring conditions.

Contribution

It provides new characterizations of when Ore extensions are PI rings under various endomorphism conditions and ring assumptions.

Findings

PI conditions for Ore extensions with injective endomorphisms

Characterization of PI Ore extensions with arbitrary endomorphisms

Results depend on ring properties like semiprimeness and noetherianity

Abstract

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.