On left derivations and related mappings

Brešar, M.; Vukman, J.

doi:10.1090/S0002-9939-1990-1028284-3

On left derivations and related mappings
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by M. Brešar and J. Vukman PDF

Proc. Amer. Math. Soc. 110 (1990), 7-16 Request permission

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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