On nilpotency of the separating ideal of a derivation

Garimella, Ramesh V.

doi:10.1090/S0002-9939-1993-1107920-X

On nilpotency of the separating ideal of a derivation
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by Ramesh V. Garimella PDF

Proc. Amer. Math. Soc. 117 (1993), 167-174 Request permission

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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