I've come to think about this problem when reading a proof in Commutative Algebra by N. Bourbaki. Say, let $R$ be a commutative ring, given 3 $R-$modules $A$, $B$, $C$, and the $R$-homomorphism $f:B \to C$. Is the following equivalent?
$f: B \to C$ is an isomorphism.
$1_A \otimes f: A \otimes B \to A \otimes C$ is an isomorphism.
I think they are equivalent, as I see the author using this fact in the proof. $1 \Rightarrow 2$ is straight-forward. But I fail to see how to prove: $2 \Rightarrow 1$. Is it correct? Any hints would be appreciated.
Thank you guys very much,
And have a good day,