Abstract: Let \(\mathscr{R}\) be a prime ring with the extended centroid \(\mathscr{C}\) and the Matrindale quotient ring \(\mathscr{Q}\). An additive mapping \(\mathscr{F}:\mathscr{R}\rightarrow \mathscr{R}\) is called a semiderivation associated with a mapping \(\mathscr{G}: \mathscr{R}\rightarrow \mathscr{R}\), whenever \( \mathscr{F}(xy)=\mathscr{F}(x)\mathscr{G}(y)+x\mathscr{F}(y)= \mathscr{F}(x)y+ \mathscr{G}(x)\mathscr{F}(y) \) and \( \mathscr{F}(\mathscr{G}(x))= \mathscr{G}(\mathscr{F}(x))\) holds for all \(x, y \in \mathscr{R}\). In this manuscript, we investigate and describe the structure of a prime ring \(\mathscr{R}\) which satisfies \(\mathscr{F}(x^m\circ y^n)\in \mathscr{Z(R)}\) for all \(x, y \in \mathscr{R}\), where \(m,n \in \mathbb{Z}^+\) and \(\mathscr{F}:\mathscr{R}\rightarrow \mathscr{R}\) is a semiderivation with an~automorphism \(\xi\) of \(\mathscr{R}\). Further, as an application of our ring theoretic results, we discussed the nature of \(\mathscr{C}^*\)-algebras. To be more specific, we obtain for any primitive \(\mathscr{C}^*\)-algebra \(\mathscr{A}\). If an anti-automorphism \( \zeta: \mathscr{A} \to \mathscr{A}\) satisfies the relation \((x^n)^\zeta+x^{n*}\in \mathscr{Z}(\mathscr{A})\) for every \({x,y}\in \mathscr{A},\) then \(\mathscr{A}\) is \(\mathscr{C}^{*}-\mathscr{W}_{4}\)-algebra, i.e., \(\mathscr{A}\) satisfies the standard identity \(\mathscr{W}_4(a_1,a_2,a_3,a_4)=0\) for all \(a_1,a_2,a_3,a_4\in \mathscr{A}\).
Keywords: prime ring, automorphism, semiderivation
For citation: Raza, M. A. and Rehman, N. A Note on Semiderivations in Prime Rings and \(\mathscr{C}^*\)-Algebras, Vladikavkaz Math. J., 2021, vol. 23, no. 2, pp. 70-77.
DOI 10.46698/d4945-5026-4001-v
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