MMN-4001

Let $\mathcal{R}$ be a prime ring, $\mathcal{I}$ be a nonzero ideal of $\mathcal{R}$, $Q$ be its maximal right ring of quotients and $C$ be its extended centroid. The aim of this paper is to show that if $\mathcal{R}$ admits a nonzero $b$-generalized derivation $\mathcal{F}$ such that $[\mathcal{F}(x^m)x^n+x^n\mathcal{F}(x^m), x^r]_k=0$ for all $x\in \mathcal{I}$, where $m, n, r, k$ are fixed positive integers, then there exists $\lambda\in C$ such that $\mathcal{F}(x)=\lambda x$ unless $\mathcal{R}\cong M_2(GF(2))$, the $2\times 2$ matrix ring over the Galois field $GF(2)$ of two elements.

Vol. 24 (2023), No. 2, pp. 819-828

DOI: 10.18514/MMN.2023.4001

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