On left derivations and related mappings
On left derivations and related mappings
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DOI：
10.1090/s0002-9939-1990-1028284-3
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发表时间：
1990
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通讯作者：
M. Brešar;J. Vukman
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文献类型：
--
作者：
M. Brešar;J. Vukman
设 R 为环，X 为左 R 模。本文的目的是研究满足 D, (ab) = aD1 (b) + bD, (a), a, b E R （左导数）和 D2(a2) = 2aD2(a), a E R （Jordan 左导数）的加性映射 D1: R -* X 和 D2: R -X。我们通过相当弱的假设证明，R 到 X 的非零乔丹左导数的存在意味着 R 是可交换的。该结果用于证明经典 Singer-Wermer 定理的两个非交换扩展。
Let R be a ring and X be a left R-module. The purpose of this paper is to investigate additive mappings D1: R -* X and D2: R -X that satisfy D, (ab) = aD1 (b) + bD, (a), a, b E R (left derivation) and D2(a2) = 2aD2(a), a E 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.