Commutativity of rings and near-rings with generalized derivations. (Q384865)

Let \(N\) be a 3-prime near-ring, and let \(f\) and \(g\) be nonzero generalized derivations on \(N\). Let \(V\) be a nonzero semigroup ideal of \(N\) -- i.e. a subset such that \(VN\subseteq V\) and \(NV\subseteq V\); and let \(U\) be a nonempty subset of \(N\). The authors explore the commutativity results which follow from the following hypotheses: (i) \([f(V),V]=\{0\}\); (ii) \(f(uv)=f(vu)\) for all \(u\in U\) and \(v\in V\); (iii) \([f(U),g(V)]=\{0\}\), where \(U\) and \(V\) are both semigroup ideals; (iv) \(f(uv)=-f(vu)\) for all \(u\in U\) and \(v\in V\); (v) \(f(v)x=-xf(v)\) for all \(v\in V\) and \(x\in N\). Some of the results are proved only for prime rings.