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If $A$ is a Noetherian local ring with maximal ideal $\mathfrak m$, how do you show that $\mathfrak m^{i}/\mathfrak m^{i+1}$ is a finitely-generated $A/\mathfrak m$-module/vector space?

I know each $\mathfrak m^i$ (and each $\mathfrak m^{i}/\mathfrak m^{i+1}$) is a f.g. $A$-module...

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    As a comment, you can generalize the statement. It hold for any ring $A$, and a maximal ideal $m$ that is finitely generated. Note that this follows because if an ideal $I$ is finitely generated over a ring, then all powers of $I$ are also finitely generated over that ring.2012-11-01

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