# Analogue of Krull intersection theorem for Symbolic powers of ideals

All rings below are commutative with unity and Noetherian.

Let $$R$$ be a domain or a local ring and $$J$$ be a proper ideal. Is it true that $$\bigcap_{n>1} J^{(n)}=(0)$$ ? If this is not true under these conditions, what conditions are known to make it true ?

($$J^{(n)} := \bigcap_{P \in \mathrm{Ass}(R/J)} \phi_P^{-1}(J^nR_P)$$ is the $$n$$th symbolic power of $$J$$, where $$\phi_P : R \to R_P$$ is the localisation map. )

UPDATE: As shown by Mohan's simple, beautiful answer below, the answer is yes when $$R$$ is a domain. So my question now is only about local rings.

• This thread should clarify you what's going on for $J$ a prime ideal, and why the intersection is not necessarily equal to $(0)$ in the local case, and why it is equal to $(0)$ when $R$ is a domain. – user26857 Apr 2 at 14:54

If $$x\in \cap J^{(n)}$$, then $$\phi_P(x)\in \cap J^n R_P=0$$. But $$\phi_P$$ is injective if $$R$$ is a domain.