A prime is minimal among primes containing an ideal Let $I$ be an ideal in a noetherian ring $R$, and $P$ prime containing $I$. I must prove that if in the localization $R_P$, $R_P/I_P$ is annihilated by a power of $P_P$, then $P$ is minimal among primes containing $I$. I have a proof by the book of D. Eisenbud but there are some parts that i don't understand. Can anyone give me an explanation or a source to study? Thanks in advance.

**Proposition:** Let $(R, m)$ be a local ring. If $m$ is nilpotent, i.e. $m^n = 0$ for some $n$, then $m$ is the only prime ideal of $R$.

Proof: Let $p$ be any prime ideal of $R$, then $m^n = 0 \subseteq p \implies m \subseteq p$, so $m = p$ (since $m$ is maximal).

Now use the correspondence between primes sitting between $I$ and $p$ (i.e. $I \subseteq q \subseteq p$), and primes of $R_p/I_p$.