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One-sided ideals and derivations of prime rings. (English) Zbl 0820.16032
The author proves three results about a derivation acting on a right ideal in a prime ring. For $$R$$ a prime ring with $$\text{char }R \neq 2$$, $$U$$ a nonzero right ideal of $$R$$, and $$D$$ a nonzero derivation of $$R$$, the three theorems are: (1) the subring generated by $$D(U)$$ contains no nonzero right ideal of $$R$$ if and only if $$D(U)U = 0$$; (2) if $$E$$ is a nonzero derivation of $$R$$ then $$ED(U) = 0$$ precisely when $$E(U)U = D(U)U = 0$$, $$E=\text{ad}(a)$$, $$D=\text{ad}(b)$$, and $$aU = bU = ba = 0$$; and (3) when $$\text{char }R = 0$$, if $$D(u)^ n U = 0$$ for $$n > 1$$ fixed and each $$u \in U$$, then $$D(u)U = 0$$.

##### MSC:
 16W25 Derivations, actions of Lie algebras 16N60 Prime and semiprime associative rings 16D25 Ideals in associative algebras 16N40 Nil and nilpotent radicals, sets, ideals, associative rings
##### Keywords:
right ideals; prime rings; derivations
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