Here is the question:
If $M$ is any $A$-module, then $T(M)$ is the kernel of the mapping $x \mapsto 1 \otimes x$ of $M$ into $K \otimes_A M$ where $K$ is the field of fractions of $A$.
Here, $A$ is an integral domain. The hint is as follows:
Show that $K$ may be regarded as the direct limit of its submodules $A\xi$ for $\xi \in K$, and 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 $\xi^{-1}x = 0$.
I have managed to show that $K$ is indeed the direct limit of the submodules of the form $K_i := \frac{1}{p_i}A$ for $p_i \in A - \{0\}$. However, I have no idea how to proceed after showing this. If we have $1 \otimes x = 0$ in $K \otimes_A M = \varinjlim (K_i \otimes_A M)$, then there exists some $k_i \otimes m$ in $K_i \otimes_A M$ that maps via the $i$-th projection $\mu_i$ to $1 \otimes x = 0$, and by previous exercises this implies that $k_i \otimes m = 0$. This feels like returning to the beginning; how can I finish the proof using the hint? (Also, the direct system the hint is giving is different from mine; I can't really see how to make a direct system out of any $A\xi$.)
I am aware of the proof using the fact $S^{-1}A\otimes_A M \cong S^{-1}M$, so I would like an answer regarding the above issue.
