# Intersection of all symbolic powers of a prime ideal

This is Exercise 8.37 in R.Y.Sharp's Steps in Commutative Algebra:

Let $$P$$ be a prime ideal of the commutative Noetherian ring $$R$$. Prove that $$\bigcap_{n=1}^{\infty} P^{(n)}=\{ r\in R \mid \exists s\in R \setminus P ,sr=0 \}$$ in which $$P^{(n)}=(P^n)^{\text{ec}}$$ with extension and contraction notation in conjunction with the natural ring homomorphism $$R\to R_P$$.

There's 2 reasons I got stuck. First, I'm confused with $$\bigcap_1^{\infty}$$, I don't know how to use this notation. Second, the only theorem I know involving with $$\bigcap_1^{\infty}$$ is Krull's Intersection Theorem, stating "If $$I\subseteq\mathrm{Jac}(R)$$ then $$\bigcap_{n=1}^{\infty}I^n=0$$", which I believe is useless in this situation.

So help me with this problem. THank you.

• Your guess that the KIT is useless is wrong. If $x\in P^{(n)}$ then $x\in P^nR_P=(PR_P)^n$. Can you take it from here? – user26857 Jan 12 '17 at 15:18
• Okay, thanks to your hint, I solved the inclusion. One more thing, now that I recheck, I was wrong about the converse. Now I solved it using your hint, too. – chí trung châu Jan 12 '17 at 15:38
• like this, $r/1=0\in (P^n)^e$, so $r\in (P^n)^{ec}$. It's true, right? – chí trung châu Jan 12 '17 at 15:39
• It sounds right. – user26857 Jan 12 '17 at 15:43
• Ok. Thank you for your help @user26857 – chí trung châu Jan 12 '17 at 15:44

• $\bigcap_{n=1}^{\infty} P^{(n)}=\bigcap_{n=1}^{\infty} (P^{n})^{ec}= (\bigcap_{n=1}^{\infty} (P^{e})^{n})^{c}=(\frac{0}{1})^{c}$
• $(\frac{0}{1})^{c}=\{r\in R$: $\exists s\notin P$ such that $sr=0\}$