Let $R$ be a ring, and let $D$ be a multiplicatively closed subset of $R$. Is it the case that if $D^{-1}R$ is a field, then $R$ must be an integral domain?
2026-03-25 15:51:51.1774453911
If the localization of a ring is a field, then the ring is an integral domain?
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This is not true, one reason being that being an integral domain is not a local property. For a concrete example take $\mathbb{Z}/6\mathbb{Z}$, this is not an integral domain, but if we localise at the set $\{ 1, 3, 5\} = \mathbb{Z}/6\mathbb{Z} \setminus (2)$ we obtain a finite integral domain and hence a field. You can find some details about determining this localisation here.