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Practice problem

Commutative Algebra

By Victor Wang

Let SS be a multiplicative subset inside a ring AA. Recall that an ideal II of AA is saturated if it contains all elements x∈Ax\in A such that Sx∩I≠∅Sx\cap I\neq \emptyset (meaning sx∈Isx\in I for some s∈Ss\in S).

What are the proper ideals of the localization S−1AS^{-1}A? Describe a one-to-one correspondence, using pullback by the localization map A→S−1A.A\to S^{-1}A.

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Type a concept, like “slope” or “photosynthesis”.

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