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On nilpotency of the separating ideal of a derivation. (English) Zbl 0794.46042
Summary: We prove that the separating ideal \(S(D)\) of any derivation \(D\) on a commutative unital algebra \(B\) is nilpotent if and only if \(S(D)\cap (\bigcap\mathbb{R}^ n)\) is a nil ideal, where \(\mathbb{R}\) 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.

MSC:
46J05 General theory of commutative topological algebras
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