Derivations satisfying certain algebraic identities on Jordan ideals.
A ring \(R\) with involution * is defined to be *-prime if \(aRb=aRb^*=\{0\}\) implies \(a=0\) or \(b=0\). An additive subgroup \(J\) is called a *-Jordan ideal if \(J^*=J\) and \(xr+rx\in J\) for all \(x\in J\) and \(r\in R\). It is proved that if a 2-torsion free *-prime ring with nonzero *-Jordan ideal \(J\) admits a nonzero derivation \(d\) such that \(d([x,y])=0\) for all \(x,y\in J\) or \(d([x,y])=[x,y]\) for all \(x,y\in J\), then \(R\) is commutative.
scientific article; zbMATH DE number 6112840