On generalized semiderivations of prime near rings
Summary: Let \(N\) be a near ring. An additive mapping \(F : N \rightarrow N\) is said to be a generalized semiderivation on \(N\) if there exists a semiderivation \(d : N \rightarrow N\) associated with a function \(g : N \rightarrow N\) such that \(F(x y) = F(x) y + g(x) d(y) = d(x) g(y) + x F(y)\) and \(F(g(x)) = g(F(x))\) for all \(x, y \in N\). In this paper we prove that prime near rings satisfying identities involving semiderivations are commutative rings, thereby extending some known results on derivations, semiderivations, and generalized derivations. We also prove that there exist no nontrivial generalized semiderivations which act as a homomorphism or as an antihomomorphism on a 3-prime near ring \(N\).
scientific article; zbMATH DE number 6873269