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Proceedings of the American Mathematical Society

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On the structure of semiderivations in prime rings

Author: Chen-Lian Chuang
Journal: Proc. Amer. Math. Soc. 108 (1990), 867-869
MSC: Primary 16A72
MathSciNet review: 1002154
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Abstract: Let $ R$ be a prime ring. By a semiderivation associated with a function $ g:R \to R$, we mean an additive mapping $ f:R \to R$ such that, for all $ x,y \in R,f(xy) = f(x)g(y) + xf(y) = f(x)y + g(x)f(y)$ and $ f(g(x)) = g(f(x))$. It is known that $ g$ must necessarily be a ring endomorphism. Here it is shown that $ f$ must be an ordinary derivation or of the form $ f(x) = \lambda (x - g(x))$ for all $ x \in R$, where $ \lambda $ is an element of the extended centroid of $ R$.

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Keywords: Semiderivation, derivation, endomorphism, extended centroid
Article copyright: © Copyright 1990 American Mathematical Society

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