All rings below are commutative, with unity and Noetherian.
For a prime ideal $P$ in $R$, if $\phi_P : R \to R_P$ is the localisation map, and $I$ is an ideal in $R_P$, let us write $I \cap R$ to mean $\phi_P^{-1} (I)$.
For an ideal $J$ in $R$ and integer $n\ge 1$ , let us define $J^{\{n\}} := \cap_{P \in Min Ass (R/J)} (J^nR_P \cap R)$ ; where $Min Ass (R/J)$ is just the collection of minimal primes over $J$.
$J^{[n]} := \cap_{J \subseteq P} (J^nR_P \cap R)$ and let $J^{(n)}$ be the usual symbolic power defined as
$J^{(n)} := \cap_{P \in Ass (R/J)} (J^nR_P \cap R)$.
Obviously we have the chain : $ J^{[n]} \subseteq J^{(n)} \subseteq J^{\{n\}}$ .
Is there any nice conditions on $R$ or $J$ which makes any of the above inclusions an equality ? Also , I would like to know, does the other two powers I have defined, apart from the symbolic power, exist in literature ?