Action of n-derivations and n-multipliers on ideals of (semi)-prime rings

Abstract

The present paper aims to investigate the containment of nonzero central ideal in a ring $ \mathcal{R} $ when the trace of symmetric $ n $-derivations satisfies some differential identities. Lastly, we prove that in a prime ring $ \mathcal{R} $ of suitable torsion restriction, if $ \mathfrak{D}, \mathcal{G} : \mathcal{R}^n \rightarrow \mathcal{R} $ are two nonzero symmetric $ n $-derivations such that $ \mathcal{f}(\vartheta)\vartheta +\vartheta\mathcal{g}(\vartheta) = 0 $ holds $ \forall \; \vartheta \in \mathcal{W} $, a nonzero left ideal of $ \mathcal{R} $ where $ \mathcal{f} $ and $ \mathcal{g} $ are the traces of $ \mathfrak{D} $ and $ \mathcal{G} $, respectively, then either $ \mathcal{R} $ is commutative or $ \mathcal{G} $ acts as a left $ n $-multiplier. Finally, we characterize symmetric $ n $-derivations in terms of left $ n $-multipliers.

Keywords

• semiprime ring
• ideal
• derivation
• symmetric $ n $-derivation
• $ n $-multiplier