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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On nilpotency of the separating ideal of a derivation


Author: Ramesh V. Garimella
Journal: Proc. Amer. Math. Soc. 117 (1993), 167-174
MSC: Primary 46H40; Secondary 46J05
MathSciNet review: 1107920
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Abstract: We prove that the separating ideal $ S(D)$ of any derivation $ D$ on a commutative unital algebra $ B$ is nilpotent if and only if $ S(D) \cap (\bigcap {{R^n})} $ is a nil ideal, where $ R$ is the Jacobson radical of $ B$. Also we show that any derivation $ D$ on a commutative unital semiprime Banach algebra $ B$ is continuous if and only if $ \bigcap {{{(S(D))}^n} = \{ 0\} } $. Further we show that the set of all nilpotent elements of $ S(D)$ is equal to $ \bigcap {(S(D) \cap P)} $, where the intersection runs over all nonclosed prime ideals of $ B$ not containing $ S(D)$. As a consequence, we show that if a commutative unital Banach algebra has only countably many nonclosed prime ideals then the separating ideal of a derivation is nilpotent.


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DOI: https://doi.org/10.1090/S0002-9939-1993-1107920-X
Article copyright: © Copyright 1993 American Mathematical Society