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