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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
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