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.

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    $\begingroup$ 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. $\endgroup$ – user26857 Apr 2 '19 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.

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  • $\begingroup$ Ah yes, thanks a lot ... do you have any idea about the local ring case ? $\endgroup$ – user Apr 2 '19 at 13:25

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