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I have a small question, which I could not verify with a quick search on the internet. Is it true that if I have a commutative ring $A$, and some multiplicative set $S$, then the localized ring $S^{-1}A$ is always a local ring?

Ben V
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No, this is not the case if $S = \{1\}$, for example, since $S^{-1}A \approx A$ in that case. However, $S^{-1}A$ will always be local if $S = A - \mathfrak{p}$ for some prime ideal $\mathfrak{p}$.

Dave
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