Finding equidimensional radical of an ideal In the article by $\textit{Eisenbud D., Huneke C., Vasconcelos W.}$ titled as $\textit{Direct Methods for Primary Decomposition (1992, Inventiones Mathematicae,110, pg.207 - 235)}$ Proposition 2.3 is as follows:
If $J$ $\subset$ $I$ $\subset$ $S =$ $k[x_1,...x_n]$ are ideals of the same dimension, and $J$ is equidimensional with radical $J$´, then the equidimensional hull of the radical of $I$ is given by the formula:
equidimensional radical of $I = (J´:(J´:I))$.
This proposition is suggested to be proven by using last statement of Lemma 2.4 which is as follows:
If $J$ is an ideal in a Noetherian ring $R$, and $M$ is a finitely generated $R-module$, then $...$c)If $J$ is a radical ideal, then $(I:J)$ is radical and $(I:J)$ = $\bigcap P_j$ where $P_j$ ranges over all primes containing $I$ but not containing $J$.
To prove Prop.2.3, I have $(J´:(J´:I)) = (J´: \bigcap R_i) $ where $R_i$ are prime ideals such that $J´ \subset R_i$ but $I \not\subset R_i$. Now, if  $J´ \subset R_i$ then $R_i$ is one of the prime ideals containing $J$, by definition of radical ideal (which is intersection of all prime ideals containing the ideal in concern). So, since dimensions of $I$ and $J $ are equal and $J \subset J´ \subset R_i $, therefore $R_i$ should contain $I$, but lemma 2.4 above says the opposite. What do I miss?
Thanks in advance!

My supervisor's answer: $R_i$ need not contain $I$ since it does not have to be the maximal prime ideal chain that determines the dimension of the ideal $J$.
