Let $R$ be a commutative ring, and let $S\subseteq R$ be a subring. Consider a prime ideal $\mathfrak p\subseteq R$ and let $\mathfrak q=\mathfrak p\cap S$ be the restricted prime ideal in $S$. When does the embedding
$$ S\hookrightarrow R$$
induce an embedding
$$S_\mathfrak q\hookrightarrow R_\mathfrak p?$$
If this is not true in general, under what assumptions does it hold?
By the exactness of localization, we have an embedding $S_\mathfrak q\hookrightarrow S_\mathfrak q\cdot R$, but I don't if this can be extended to $S_\mathfrak qR\to R_\mathfrak p$.
Best Answer
About your first question, with $R=k[x,y]/(xy), p=(x),S=k[x],q=(x)$,
Since $y$ becomes a unit in $R_p$ then $x$ is in the kernel of $R\to R_p$ ie. $R_p=Frac(k[x,y]/(x))$,
Whereas $S_q = k[x]_{(x)}$.