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Generalized derivations of left faithful rings. (English) Zbl 0946.16026

Let \(R\) be a left faithful ring, \(U\) its right Utumi quotient ring, and \(T\) a dense right ideal of \(R\). A generalized derivation is an additive \(g\colon T\to U\) satisfying \(g(xy)=g(x)y+xD(y)\) for some derivation \(D\colon T\to U\) and all \(x,y\in T\). The author proves that \(D\) extends to a derivation of \(U\) and that \(g\) extends to a generalized derivation on \(U\) so that \(g(x)=ax+D(x)\) for some \(a\in U\) and all \(x\in T\). Further, \(a\in U\) and \(D\in\text{Der}(U)\) are unique if \(R\) is a semiprime ring. This characterization is used to obtain three results known for derivations with nilpotent values. Namely, let \(n>0\) be fixed, \(g\colon T\to U\) a generalized derivation, and \(g(x)^n=0\) for all \(x\in A\subseteq R\). Then \(g=0\) if \(R\) is semiprime and \(A=T\) or if \(R\) is prime and \(A\) is a noncommutative Lie ideal of \(R\), and \(g(A)A=0\) if \(R\) is prime and \(A\) is a nonzero right ideal of \(R\).

MSC:

16W25 Derivations, actions of Lie algebras
16N60 Prime and semiprime associative rings
16W10 Rings with involution; Lie, Jordan and other nonassociative structures
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References:

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