3
$\begingroup$

Let $R$ be a commutative unital ring and let $M$ be an $R$-module and let $S$ be a multiplicative subset of $R$.

Today I proved both of the following: $ S^{-1} R\otimes_R S^{-1}M \cong S^{-1} M$ and $ S^{-1} R \otimes M \cong S^{-1} M$

Now I'm slightly confused.

Either my proofs are wrong or $C \otimes A \cong C \otimes B$ does not imply $A \cong B$. But I can't come up with an example. Can someone give me an example? (Or tell me that my proofs are wrong.)

  • 0
    $(\mathbb Z/5\mathbb Z) \otimes (\mathbb Z/3\mathbb Z) = (\mathbb Z/7\mathbb Z )\otimes (\mathbb Z/3\mathbb Z ) =0$ but right sides are not isomorphic (everything as $\mathbb Z$-module).2014-09-09

1 Answers 1

8

You are correct in noticing that tensoring with a fixed module isn't an injective operation. Various things can go wrong; probably the simplest thing to notice is that tensoring can kill torsion. To repeat the example given in the comments, both $\mathbb{Z}_2\otimes \mathbb{Q}$ and $\mathbb{Z}_2\otimes\mathbb{R}$ (everything in sight is a $\mathbb{Z}$-module) are trivial, while $\mathbb{Q}$ and $\mathbb{R}$ aren't isomorphic. There is a multitude of examples in the same vein, e.g. tensoring any two finitely generated $\mathbb{Z}$-modules of the same rank with $\mathbb{Q}$ will produce isomorphic modules.