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?