Lie ideals and Jordan ideals in \(3\)-prime near-rings with derivations. (Q2794775)

Let \(N\) be a zero-symmetric near-ring with multiplicative center \(Z(N)\). Define a Jordan (resp. Lie) ideal of \(N\) to be an additive subgroup \(J\) (resp. \(U\)) such that \(jx+xj\) and \(xj+jx\) are in \(J\) for all \(j\in J\) and \(x\in N\) (resp. \(ux-xu\in U\) for all \(u\in U\) and \(x\in N\)).NEWLINENEWLINE The paper extends to near-rings results for prime rings admitting nonzero derivations satisfying certain conditions on Lie or Jordan ideals. -- Two of the results are (a) If \(N\) is 3-prime and 2-torsion free and admits a nonzero derivation \(d\) such that \(d(U)\subseteq Z(N)\), where \(U\) is a nonzero Lie ideal, then \(U\subseteq Z(N)\) and \((N,+)\) is Abelian; (b) If \(N\) is 3-prime and 2-torsion-free and \(J\) is a nonzero Jordan ideal, and \(N\) admits a nonzero generalized derivation \(F\) with associated derivation \(d\) such that \(F\) is either the zero map or the identity map on \(J\), then \(d=0\) or \(J\) is multiplicatively commutative.