An identity involving automorphisms of prime rings inspired by Posner's theorem

Abstract

Let ${\mathcal R}$ be a prime ring with centre ${\mathcal Z}(\mathcal {R})$, $\mathcal {L}$ a non-zero Lie ideal of ${\mathcal R}$, and σ a non-trivial automorphism of ${\mathcal R}$ such that $[[\sigma (u),u], \sigma (u)] \in \mathcal {Z}(\mathcal {R})$ for all $u \in {\mathcal L}$. If $char (R) \neq 2$, then it is shown that ${\mathcal R}$ satisfies $s_4$, the standard identity in four variables.

Keywords

• Prime ring
• Lie ideal
• automorphism
• polynomial identity
• centralizing mapping