Semiderivations satisfying certain algebraic identities on prime near-rings. (Q2868586)

Let \(N\) denote a near-ring with multiplicative center \(Z\). A semiderivation on \(N\) is an additive map \(d\) for which there exists an additive map \(g\) on \(N\) such that NEWLINE\[NEWLINEd(xy)=xd(y)+d(x)g(y)=g(x)d(y)+d(x)\quad\text{ and }\quad d(g(x))=g(d(x))\quad\text{ for all }x,y\in N.NEWLINE\]NEWLINE It is proved that a 2-torsion free 3-prime near-ring \(N\) must be a commutative ring if it admits a nonzero semiderivation \(d\) satisfying one of the following: (a) \(d(N)\subseteq Z\); (b) \(d(xy)=d(yx)\) for all \(x,y\in N\).