Semiderivations satisfying certain algebraic identities on Jordan ideals. (Q373470)

The authors consider semiderivations that fix certain commutators \([x,y]=xy-yx\) in a ring \(R\) with involution *. Let \(R\) be a noncommutative, 2-torsion free, *-prime ring with nonzero Jordan ideal \(J=J^*\). A semiderivation of \(R\) is a pair \((d,g)\) where \(d,g\in\text{End}(R,+)\) commute and satisfy \(d(xy)=d(x)g(y)+xd(y)=d(x)y+g(x)d(y)\) for all \(x,y\in R\). The main result of the authors is that when \(d([x,y])=[x,y]\) for all \(x,y\in J\) then for all \(r\in R\), \(d(r)=r-g(r)\). One consequence is that the same conclusion holds, and also \(g([R,R])=0\), when \(R\) is any 2-torsion free, semiprime ring, and \(J\) is replaced by a nonzero ideal \(I\) of \(R\). The authors introduce the notion of a generalized semiderivation \((F,d,g)\) in the usual way, and they show that when \(R\) is a prime ring then either \(g=\mathrm{id}_R\) and so \(F\) is a generalized derivation, or \(F-d=c\cdot\mathrm{id}_R\) for some \(c\) in the extended centroid of \(R\).