what can be said about $spec(R_m)$, where $R_m$ is localization of $R$ at maximal ideal $m$

Question

I've seen how if $p$ is a prime ideal of $R$ and $R_p$ is the localization of $R$ at $P$, then $P_p$ is the unique maximal ideal of $R_p$, but what if we had a maximal ideal $m$ of $R$, then $R_m$ would also have a unique maximal ideal, i.e $m_m$, since all maximal ideals are prime. What else can I say about $R_m$, is there anything special about the rest of the prime ideals in $R_m$.

Answer 1

There's nothing special about the prime ideals in $R_m$ except that they correspond to the ones in $R$ that are contained in $m$. If $R$ is an integral domain then I guess that's a one-to-one correspondence.

So if $R=\mathbb C[x,y]$ the max ideals correspond to points in the $x$-$y$ plane and the non-maximal prime ideals are given by the irreducible curves, and the zero ideal. If $p$ is a point in the plane with max ideal $m_p$ then $R_{m_p}$ has prime ideals exactly given by the irreducible polynomials for which $p$ is a zero.