Let [itex]M[/itex] be a module over the commutative ring [itex]K[/itex] with unit 1. I want to prove that [itex]M \cong M \otimes K.[/itex] Define [itex]\phi:M \rightarrow M \otimes K[/itex] by [itex]\phi(m)=m \otimes 1.[/itex] This is a morphism because the tensor product is K-linear in the first slot. It is also easy to show that the map is surjective. This is where I get stuck. Suppose [itex]\phi(m)=\phi(n),[/itex] so that [itex]0 = m \otimes 1 - n \otimes 1 = (m-n) \otimes 1.[/itex] How do I prove that this implies that [itex]m=n[/itex] and thus the map is injective? More generally, how can you tell when a tensor is 0?