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Let $R$ be a Noetherian ring and let $=(h_1, ..., h_n)$ be a prime ideal in $R$. Suppose that $R$ is a finitely generated $A$-module, $A=k[h_1, ..., h_n]$. What is the Krull dimension of $R_$ as a module over itself? Does the condition "$R$ is a finitely generated $A$-module" make any difference?

Thank you.

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    @user26857 Sorry, but $p$ must be prime for what? – 2017-02-20
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    Also, if $p$ is prime, dim$R_p$ = height $p$. If $p$ is not prime, can we say that dim$R_p \leq$ height $p$? I mean can the equality hold even if $p$ is not prime? – 2017-02-20
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    @user26857 So, we cannot localize at non-prime ideal – 2017-02-20
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    @user26857 Thank you. – 2017-02-20

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