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Suppose $R$ is noetherian, suppose $M$ is a finitely generated $R$-module of grade 0. This means that there are no regular elements in his annihilator. Does this implies that $\mathrm{dim}\;R/\mathrm{Ann}\;M=\mathrm{dim}\;R$? if not what can I say about the dimension of $M$?

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Of course the equality is wrong! Take $(R,m)$ local and not CM of dimension $1$ and $M=R/m$. –  user26857 Nov 10 '12 at 10:43
    
yeah, I thought about it tonight and I came with the same example... there is something in general that we can say about the dimension of $M$? –  Chris Nov 10 '12 at 19:25
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