Existence of derivations on near-rings. (Q2843878)

This paper provides welcome information on derivations on near-rings which are not rings. The authors begin by showing that a near-ring \(N\) admits a multiplicative derivation \(d\) (i.e., a map \(d\colon N\to N\) such that \(d(xy)=xd(y)+d(x)y\) for all \(x,y\in N\)) if and only if \(N\) is zero-symmetric. They discuss existence of inner derivations on zero-symmetric near-rings satisfying various distributivity conditions, and they give several examples of nontrivial derivations on other near-rings. It is an open question whether a \(3\)-prime near-ring which is not a ring can admit a nonzero derivation, but the authors provide an example of a nonzero \((1,\sigma)\)-derivation on such a near-ring.