Assume $A$ is a ring, and $B \subset A$ is a multiplicative subset. Can the prime ideals of the localization $B^{-1}A$ be identified with those prime ideals of $A$ containing no elements of $B$?
Can prime ideals of $B^{-1}A$ be identified with prime ideals of $A$ containing no elements of $B$?
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abstract-algebra
ring-theory
localization
1 Answers
1
Yes. Let $\iota :A \to B^{-1}A$ be the localization map. Then $\iota^{-1}: \operatorname{Spec} B^{-1}A \to \operatorname{Spec}A$ is injective with the image being precisely those prime ideals of $A$ that do not meet $B$.