I think it will be obvious for some people but I started reading Algebraic Number Theory of A. Frolich & J.W.S Cassels and I don't understand the proof of a lemma. If we take $R$ a Dedekind ring, $K=\operatorname{Frac}(R)$, $U$ a vector space of finite dimension over $K$ and $T$ a $R$-submodule of $U$ then the lemma says that $$\bigcap_p T_p =T$$ where $p$ are prime ideals of R, $T_p=TR_p$ where $R_p=(R\setminus p)^{-1}R$. And in the book it is said that this is clear that : $$T \subset \cap_p T_p$$
I don't understand why :/.
Because $R$ is an integral domain, we have $R \subset R_p$ when we consider both as subsets of $K$. (Because elements of $R_p$ are of the form $\frac{r}{s}$ with $r \in R$ and $s \in R\setminus P$, and we can always set $s=1$, to get $R \ni r=\frac{r}{1} \in R_p)$
Then $T=TR \subset TR_p=T_p$, where the first equality holds because $T$ is an $R$-module. Because this holds for any $p$, we have $T \subset \cap_p T_p$