On Jordan nilpotency of index n in group rings

Abstract

Let $RG$ be the group ring of an arbitrary group $G$ over an associative and commutative ring $R$ with identity. In this paper, we provide necessary and sufficient conditions for $RG$ to be Jordan nilpotent of index $n \geq 2$. In particular, we establish a connection between the study of Jordan nilpotency in group rings and the study of Lie nilpotency in these algebras. We also classify the Jordan nilpotency of $RG$, over rings of arbitrary characteristic, up to the index $6$.