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.)