I have been thinking about this question alot but couldn't figure out an answer.
Is intersection of two local rings again a local ring?
I can't find any counterexample. So any help will be appreciated. Thanks
2026-03-25 07:31:55.1774423915
Intersection of two local rings
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The Intersection of two Local rings should not necessarily be local again. One can show that every integral domain $R$ is given as the intersection $\bigcap R_{\mathfrak{m}}$ of all its localizations at maximal ideals (considered as subrings of the field of fraction). By that you can at least see that this does not hold for general intersections. To get your example for two local rings, we can try to find an integral domain with exactly 2 maximal ideals. Choose two valuation rings of a field $K$ not containing each other and consider their intersection. This yields an integral domain with two maximal ideals (or more generally one with $n$ maximal ideals). The latter result can be found in the book multiplicative ideal theory by Gilmer (Theorem 22.8).