On \(b\)-generalized derivations in prime rings (Q6104178)

Let \(R\) be a prime ring with its Utumi quotient ring \(U\), the extended centroid \(C\), and \(Z(R)\) the center of \(R\) and \(f(x_{1}, \cdots , x_n)\) be a multilinear polynomial over \(C\). It may be noted that the extended centroid \(C\) of a prime ring \(R\) is always a field and \(C=Z(U)\). An additive mapping \(d:R \to R\) is said to be a derivation of \(R\) if \(d(xy)=d(x)y+xd(y)\) holds for all \(x,y \in R\). An additive mapping \(G:R \to R\) is called a generalized derivation of \(R\) if there exists a derivation \(d: R\to R\) such that \(G(xy)=G(x)y+xd(y)\) holds for all \(x,y\in R\). An additive mapping \(F : R \to U\) is called \(b\)-generalized derivation of \(R\) if \(F(xy) = F(x)y + bxd(y)\) for all \(x, y \in R\), where \(d: R \to U\) is an additive map. In the paper under review the author obtains a complete description of \(b\)-generalized derivation on a multilinear polynomial of a prime ring \(R\), that is, it satisfies the property \[ puF(u) + F(u)uq = G(u^2),\quad p + q \not\in C ~\text{for all} ~~u \in f(R) ~\text{for} ~~p, q \in R \] where \(F, G\) are non zero \(b\)-generalized derivations and is a multilinear polynomial over \(R\). Then for all \(x \in R\) one of the following holds: \begin{itemize} \item[1.] There exists \(a \in C, c \in U\) for which \(F(x) = ax, G(x) = xc\) with \(pa = c - aq \in C, p \in C\). \item[2.] There exists \(a \in C , c, c^{\prime} \in U\) for which \(F(x) = ax , G(x) = cx + xc^{\prime}\) with \(pa - c = c^{\prime} - qa \in C\). \item[3.] There exists \(a \in C, c \in U\) for which \(F(x) = ax, G(x) = cx\) with \((p + q)a = c\) and \(q \in C\). \item[4.] There exists \(a \in C, c \in U\) for which \(F(x) = ax, G(x) = cx\) with \(aq = c - ap \in C\). \item[5.] There exists \(a \in C , c \in U\) for which \(F(x) = ax , G(x) = cx + xc^{\prime}\) with \(pa - c = c^{\prime} - qa \in C\). \item[6.] There exists \(a \in C, c \in U\) for which \(F(x) = ax , G(x) = cx\) with \(p \in C, ap \in C, ap - c = -aq \in C\). \item[7.] \(f(x_1, \cdots, x_n)^{2}\) is central valued on R and one of the following holds: \begin{itemize} \item[(i)] There exists \(a, c \in U\) for which \(F(x) = ax, G(x)=[c, x] + xc^{\prime}\) with \(p \in C, a(p + q) = c^{\prime}\). \item[(ii)] There exists \(a, c, c^{\prime} \in U\) for which \(F(x) = xa, G(x)=[c, x] + xc^{\prime}\) with \(( p + q)a = c^{\prime}\) and \(aq \in C, q \in C\). \item[(iii)] There exists \(a \in C, c, b, v \in U\) for which \(F(x) = ax, G(x) = cx + bxv\) with \(c + bv = a(p + q)\). \item[(iv)] There exists \(a, c, b, v \in U\) for which \(F(x) = ax, G(x) = cx + bxv\) with \(ap \in C, p \in C\) and \(c + bv = a( p + q)\). \item[(v)] There exists \(a \in C, c, c^{\prime} \in U\) for which \(F(x) = ax , G(x) = cx + xc^{\prime}\) with \(( p + q)a = c + c\). \end{itemize} \end{itemize} The result is obtained by using the theory of generalized polynomial identities [\textit{K. I. Beidar} et al., Rings with generalized identities. New York, NY: Marcel Dekker (1996; Zbl 0847.16001)] and the theory of differential identities [\textit{V. K. Kharchenko}, Algebra Logic 17, 155--168 (1979; Zbl 0423.16011); translation from Algebra Logika 17, 220--238 (1978)].