# Problem Set 3

:::{.problem title="AM 3.1"}
Let $S$ be a multiplicatively closed subset of a ring $A$, and let $M$ be a finitely generated $A$-module. Prove that $S^{-1} M=0$ if and only if there exists $s \in S$ such that $s M=0$.
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:::{.problem title="AM 3.4"}
Let $f: A \rightarrow B$ be a homomorphism of rings and let $S$ be a multiplicatively closed subset of $A$. Let $T=f(S)$. Show that $S^{-1} B$ and $T^{-1} B$ are isomorphic as $S^{-1} A$-modules.

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:::{.problem title="AM 3.12"}
Let $A$ be an integral domain and $M$ an $A$-module. An element $x \in M$ is a torsion element of $M$ if Ann $(x) \neq 0$, that is if $x$ is killed by some non-zero element of $A$. Show that the torsion elements of $M$ form a submodule of $M$. This submodule is called the torsion submodule of $M$ and is denoted by $T(M)$. If $T(M)=0$, the module $M$ is said to be torsion-free. Show that

i) If $M$ is any $A$-module, then $M / T(M)$ is torsion-free.

ii) If $f: M \rightarrow N$ is a module homomorphism, then $f(T(M)) \subseteq T(N)$.

iii) If $0 \rightarrow M^{\prime} \rightarrow M \rightarrow M^{\prime \prime}$ is an exact sequence, then the sequence $0 \rightarrow T\left(M^{\prime}\right)$ $\rightarrow T(M) \rightarrow T\left(M^{\prime \prime}\right)$ is exact.

iv) If $M$ is any $A$-module, then $T(M)$ is the kernel of the mapping $x \mapsto 1 \otimes x$ of $M$ into $K \otimes_{1} M$, where $K$ is the field of fractions of $A$.

> For iv), show that $K$ may be regarded as the direct limit of its submodules $A \xi(\xi \in K)$; using Chapter 1, Exercise 15 and Exercise 20, show that if $1 \otimes x=0$ in $K \otimes M$ then $1 \otimes x=0$ in $A \xi \otimes M$ for some $\xi \neq 0$. Deduce that $\xi^{-1} x=0$.

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:::{.problem title="Ex. 1"}
Show that $\mathbb{Q}$ is a flat $\mathbb{Z}$-module which is not free
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:::{.problem title="Ex. 2"}
Prove that if $B, C$ are $A$ algebras, the tensor product algebra $B \otimes_{A} C$ has the following universal property: an algebra homomorphism $B \otimes_{A} C \rightarrow S$ is the same as a pair of algebra homomorphisms $B \rightarrow S, C \rightarrow S$.
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