On derivations in near-rings and rings. (English) Zbl 0810.16042

Bell, Howard E.; Mason, Gordon

Math. J. Okayama Univ. 34, 135-144 (1992).

In a near-ring \(R\) a derivation \(D\) is called an scp-derivation if \([x,y] = [D(x), D(y)]\), a Daif 1(2)-derivation if \(D(xy) - D(yx) = [x, y] (=[-x, y])\) (\(\forall x, y \in R\)). Various commutativity (and distributivity) results linked to such derivations are given: e.g. if \(R\) is a prime ring having a nonzero right ideal \(U\) and a derivation \(D\) such that \(\forall x,y \in U\) \([x, y] = [D(x), D(y)]\) then \(R\) is commutative.

Reviewer: C.Cotti-Ferrero (Parma)

MSC:

16Y30	Near-rings
16W25	Derivations, actions of Lie algebras
16U70	Center, normalizer (invariant elements) (associative rings and algebras)
16U80	Generalizations of commutativity (associative rings and algebras)
16N60	Prime and semiprime associative rings

Keywords:

scp-derivation; Daif 1(2)-derivation; commutativity; distributivity; derivations; prime ring