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Two sided \(\alpha \)-derivations in 3-prime near-rings. (English) Zbl 1353.16018

Let \(N\) be a zero-symmetric left near-ring with multiplicative center \(Z(N)\), and let \(\alpha\) be a mapping from \(N\) to \(N\). An additive map \(d:N\to N\) is called a \((1,\alpha)\)-derivation (resp. \((\alpha,1)\)-derivation) if \(d(xy)= d(x)y+\alpha(x)\, d(y)\) (resp. \(d(xy)= d(x)\,\alpha(y)+ xd(y))\) for all \(x,y\in N\); and \(d\) is called a two-sided \(\alpha\)-derivation if it is both \((1,\alpha)\)-derivation and an \((\alpha,1)\)-derivation.
The authors generalize some commutativity results which were known for derivations (which are \((1,1)\)-derivations). The following are among the theorems proved:
(a) If \(N\) is 3-prime and \(d\) is a nonzero \((1,\alpha)\)-derivation for a homomorphism \(\alpha\), then \(N\) must be a commutative ring if either \(d(N)\subseteq Z(N)\) or \(d(xy)- yx)= 0\) for all \(x,y\in N\).
(b) If \(N\) is 2-torsion-free and 3-prime, then \(N\) admits no nonzero two-sided \(\alpha\)-derivation \(d\) such that \(d(xy+ yx)= 0\) for all \(x,y\in N\).

MSC:

16N60 Prime and semiprime associative rings
16W25 Derivations, actions of Lie algebras
16Y30 Near-rings
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References:

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