Basic Ring Theory - 17

Groups, Rings, Domains, Modules, etc, Galois theory
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Tsakanikas Nickos
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Basic Ring Theory - 17

#1

Post by Tsakanikas Nickos »

Prove the following assertion (which is relative to the one in Basic Ring Theory - 17) :

If $A$ is an integral domain, then $A$ is equal to the intersection (inside its quotient field $Q(A)$) of its localizations at all maximal ideals, i.e. \[ A = \bigcap_{ \mathfrak{m} \in \text{MaxSpec}(A) } A_{\mathfrak{m}} \]where $\text{MaxSpec}(A)$ is the set of all maximal ideals of $A$.
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