Let $R$ be a commutative ring with identity, not necessarily a domain, and let $S_i$ be a family of multiplicative closed subsets of $R$. What is the exact meaning of $\bigcap_i S_i^{-1}R$?
For example, for an integral domain $R$, it is well known that for two prime ideals $p_1$ and $p_2$ of $R$, we have $R_{p_1}\cap R_{p_2}=R_{p_1\cup p_2}$.