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Integral Domain, Localization and Maximal Ideals

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Let A be an integral domain. For a prime ideal P C A and let S = A P be the complement of P in A. Observe that S is multiplicatively closed (WHY?) and form the subring A_p := S^-1 A of the field of fractions F of A. We regard A as a subring of A_p as usual - i.e. an element a E A is identified with the fraction a/1 = as/s for each s E S.

The ring A_p is referred to as the location of A at P.

(i) Show that A_p is a local ring whose unique maximal ideal is PA_p = {a/b | a E P, b E A P}.

(ii) Show that A = np A_p

where the intersection occurs inside the field F, and the intersection is taken over all prime ideals P of A.

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