I have encountered a problem which requires me to prove that ideal inclusion is a local property. That is to say, suppose $S,T \subset R$. Show that $S \subset T \Leftrightarrow SR_P \subset TR_P$ for every prime ideal $P \subset R$.
This is rather an elementary exercise but I think it would be tedious to start with definition and prove the if and only if statement. I was wondering if there is an extremely easy and fast way to show this statement? Since I know a proposition saying that being injective or surjective as Noetherian module is a local property. The proof will be one line for Noetherian ring if we just let the injective map be inclusion. However, it is not always true for general rings. Are there any famous propositions we can use to JUST finish this proof in several sentences, or we must prove this statement from definition?
One implication is easy. For the harder one, suppose $S \not \subset T$. Fix an element $x \in S \setminus T$. Show that the set of all ideals of $R$ that contain $T$ and do not contain $x$ satisfies the condition for Zorn's lemma, so it has a maximal element. Show that such a maximal element $P$ is a prime ideal in $R$, and $SR_P \not \subset TR_P$.