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Let $R$ be a Noetherian ring and $M$ be a $R$ module.

Suppose the set of associated primes of a module $M$ is finite, then does it necessarily mean that the support of the module $M$ is closed?

Support of $M$ is defined to the set of the all primes $p$ such that $M_p\neq 0$.

dongrugose
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1 Answers1

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No. Take $R = \mathbb{C}[x]$ and $M = R_x$, the localization at $x$. Then (0) is the only associated prime of $M$, but Supp $M$ is Spec $R$ $\setminus V(x)$, which is open but not closed.

Youngsu
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