Let $R$ be an integral domain, and let $K$ be its field of fractions. Let $C$ be an $R$-module. Then is $C \cong C \otimes_R R$ injectively contained in $C \otimes_R K$?
Define $\phi: C \otimes_R R \to C \otimes_R K$ by $c \otimes r \mapsto c \otimes r$ and extend it additively. I'm having trouble showing that this map is well-defined and injective.