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Let $k$ be a field. Let $R = \prod_{n \in \mathbb{N}} k$. Due to the answer in this question Infinite product of fields, we know that $R$ is zero dimensional, and the localization $R_m$ at every maximal ideal $m \subset R$ is a field. In fact, it is $R/m$. Is $R_m = k$ or is this not true in general for an arbitrary field?

If the statement is not true in general, what is an example where it fails to be true?

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    $R$ is a commutative von Neumann regular ring, and this shows that $\dim R=0$ and $R_m$ is a field for any maximal ideal $m$. I guess that for $k=\mathbb{Q}$ and $m$ the maximal ideal containing $\oplus\mathbb{Q}$, the quotient ring $R/m$ has cardinality $c=|\mathbb{R}|$, and therefore it is not isomorphic to $k$.2012-06-21

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