Exercise 111.7.1. Let $A$ be a ring. Let $S \subset A$ be a multiplicative subset. Let $M$ be an $A$-module. Let $N \subset S^{-1}M$ be an $S^{-1}A$-submodule. Show that there exists an $A$-submodule $N' \subset M$ such that $N = S^{-1}N'$. (This useful result applies in particular to ideals of $S^{-1}A$.)

## 111.7 Localization

Exercise 111.7.2. Let $A$ be a ring. Let $M$ be an $A$-module. Let $m \in M$.

Show that $I = \{ a \in A \mid am = 0\} $ is an ideal of $A$.

For a prime $\mathfrak p$ of $A$ show that the image of $m$ in $M_\mathfrak p$ is zero if and only if $I \not\subset \mathfrak p$.

Show that $m$ is zero if and only if the image of $m$ is zero in $M_\mathfrak p$ for all primes $\mathfrak p$ of $A$.

Show that $m$ is zero if and only if the image of $m$ is zero in $M_\mathfrak m$ for all maximal ideals $\mathfrak m$ of $A$.

Show that $M = 0$ if and only if $M_{\mathfrak m}$ is zero for all maximal ideals $\mathfrak m$.

Exercise 111.7.3. Find a pair $(A, f)$ where $A$ is a domain with three or more pairwise distinct primes and $f \in A$ is an element such that the principal localization $A_ f = \{ 1, f, f^2, \ldots \} ^{-1}A$ is a field.

Exercise 111.7.4. Let $A$ be a ring. Let $M$ be a finite $A$-module. Let $S \subset A$ be a multiplicative set. Assume that $S^{-1}M = 0$. Show that there exists an $f \in S$ such that the principal localization $M_ f = \{ 1, f, f^2, \ldots \} ^{-1}M$ is zero.

Exercise 111.7.5. Give an example of a triple $(A, I, S)$ where $A$ is a ring, $0 \not= I \not= A$ is a proper nonzero ideal, and $S \subset A$ is a multiplicative subset such that $A/I \cong S^{-1}A$ as $A$-algebras.

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