If $A$ is an integral domain with a finite number of prime ideals is it possible to get the field of fractions localizing only by a set $\{a^k\}$?
2026-03-30 06:10:54.1774851054
If $A$ is an integral domain with a finite number of primes then $Q(A)=A_a$ for some $a \in A$
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The answer is yes and the proof is trivial: For any non-zero prime, take one non-zero element and let $a$ to be the product of all those elements.
The only prime ideal in $A_a$ is the zero-ideal because by our construction any other prime ideal in $A$ contains $a$, hence is the unit ideal in the localization.