On a functional equation related to derivations in prime rings.
The authors consider a functional identity for a prime ring that leads to an extension of results about when a Jordan derivation of a prime ring is actually a derivation. Let \(R\) be a ring, \(D\colon R\to R\) a nonzero additive map, integers \(n,m\geq 1\), \(E(x)=2D(x^{m+n+1})-(m+n+1)(x^mD(x)x^n+x^nD(x)x^m)\), and \(p(X)=\sum x_{\sigma(1)}\cdots x_{\sigma(m+n+1)}\) taken over all \(\sigma\) in the symmetric group \(S_{m+n+1}\). The main result of the paper takes \(R\) to be a \(K\)-algebra, \(L\) a Lie subalgebra of \(R\) so that \(L\) is \(2(m+n+1)\)-free and \(p(L^{m+n+1})\subseteq L\). If \(E(x)=0\) for all \(x\in L\) then on \(L\), \(2D(x)=cx+\rho(x)\) for some \(c\in C(L)\) and \(\rho\colon L\to C(L)\). This result is used to prove that when \(R\) is a prime ring with \(\text{char\,}R=0\) or \(\text{char\,}R\geq m+n+1\) then \(E(r)=0\) for all \(r\in R\) forces \(R\) to be commutative and \(D\) to be a derivation of \(R\).
scientific article; zbMATH DE number 6109333
scientific article