## 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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