\(b\)-generalized skew derivations on multilinear polynomials in prime rings (Q824440)

Let \(R\) be a prime ring with right Martindale quotient ring \(\mathcal {Q}_r\), \(b\in \mathcal {Q}_r\), \(d:R\to \mathcal {Q}_r\) an additive mapping, and \(\alpha \) an automorphism of \(R\). An additive mapping \(F: R\to \mathcal {Q}_r\) is called a \(b\)-generalized skew derivation with associated term \((b, \alpha, d)\) if\[F(xy)=F(x)y+b\alpha (x)d(y)\] for all \(x, y\in R\). Let \(R\) be a prime ring of characteristic different from 2 and \(C\) be its extended centroid. In this paper, the authors study all possible forms of two \(b\)-generalized skew derivations \(F\) and \(G\) satisfying the condition \[F(x)x-xG(x)=0\] for all \(x\in S\), where \(S\) is the set of the evaluations of a multilinear polynomial \(f(x_1, \dots , x_n)=0\) over \(C\) with \(n\) non-commuting variables. The authors also present several potential research topics related to this paper. For the entire collection see [Zbl 1461.16003].