I'm reading the paper A formula for the core of an ideal of C. Polini and B. Ulrich and I'm with serious trouble in a passage. Since I guess it is just only smart trick, I decided to ask here.
Some definitions:
Let $R$ be a Noetherian ring, $I$ an ideal of $R$. A subideal $J\subseteq I$ is said a reduction of $I$ if there is $n\in \mathbf{N}$ such that $JI^n = I^{n+1}$.
The problem is:
We have a Cohen-Macaulay local ring $R$ with infinite residue field $k$, $I$ an $R$-ideal with $ht(I)=\ell(I)=1$ and $J$ a minimal reduction of $I$. We know that $JI^r = I^{r+1}$. Since $ht(I)=\ell(I)=1$, we know $J$ is a principal ideal generated by a $R$-regular element. The article said that if the $char(k) = 0$ or $char(k)>r$, we have $I^r = \sum_{y \in I}(J,y)^r$.
Well, the inclusion $\supseteq$ is trivial, but I don't have any idea how to prove the other inclusion.
Thanks, any help will be welcome.