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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


On left derivations and related mappings

Authors: M. Brešar and J. Vukman
Journal: Proc. Amer. Math. Soc. 110 (1990), 7-16
MSC: Primary 16W25; Secondary 16U80, 16W10, 16W80, 46H05
MathSciNet review: 1028284
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Abstract: Let $ R$ be a ring and $ X$ be a left $ R$-module. The purpose of this paper is to investigate additive mappings $ {D_1}:R \to X$ and $ {D_2}:R \to X$ that satisfy $ {D_1}(ab) = a{D_1}(b) + b{D_1}(a),a,b \in R$ (left derivation) and $ {D_2}({a^2}) = 2a{D_2}(a),a \in R$ (Jordan left derivation). We show, by the rather weak assumptions, that the existence of a nonzero Jordan left derivation of $ R$ into $ X$ implies $ R$ is commutative. This result is used to prove two noncommutative extensions of the classical Singer-Wermer theorem.

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Additional Information

PII: S 0002-9939(1990)1028284-3
Keywords: Derivation, Jordan derivation, left derivation, Jordan left derivation, prime ring, semiprime ring, Banach algebra
Article copyright: © Copyright 1990 American Mathematical Society

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