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revised Is this module noetherian?
Made precise, per comments
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comment Is this module noetherian?
@rschwieb, I don't understand your comment, nor the downvote to the question. It is customary in commutative algebra that when one says ideal, one means proper ideal. Thus, the question as stated makes sense. Also, your "solution" does not make much sense, because if the op thought that the enveloping algebra is noetherian then the answer to both of his question will be obviously yes (because his ideal is f.g, and f.g ideals over noetherian rings are noetherian and coherent).
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revised Has this variation of Hochschild cohomology been studied?
odded operator name for the Ext functor
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suggested approved edit on Has this variation of Hochschild cohomology been studied?