I'm reading some lecture notes of Pete L. Clark, and there's one problem that I cannot solve. It's on page 45 of this book: Commutative Algebra. The problem reads as follow:
Let R be a domain with fraction field K.
1. Show that K is a uniquely divisible $R-$module. (I think I get it, but I'm not sure, isn't this the way we construct a fraction field from a domain?)
2. Let $M$ be any $R-$module. Show that the natural map $M \otimes M \otimes_R K$ is injective iff $M$ is torsion-free.
I think the author made a little typo here, it should read $M \to M \otimes_R K$, instead of $M \otimes M \otimes_R K$, does my correction look good? Or should it be corrected in another way?
This is where I have no idea, I can prove $M \to M \otimes_R K$ is injective $\Rightarrow$ $M$ is torsion-free by Proof by Contradiction. But how can I prove the other way round? I would be very glad if someone can give me a push on this.
3. Show that for any $R-$module $M$, $M \otimes_R K$ is uniquely divisible.
I can show that it's divisible, but do I need to use 2. to show that it's uniquely divisible?
4. Show that $K / R$ is divisible, but not uniquely divisible.
I get this part. :)
I think I'm having some troubles with Tensor Product, it's pretty hard to visualize what its elements are, such as I have no idea what 0 looks like. I do know that $0 \otimes k = m \otimes 0 = 0$, but that's not all the 0's in $M \otimes_R K$.
So, thanks every one,
And have a good day,