  Let $(R,\fm)$ be a Cohen-Macaulay local ring of positive dimension $d$ and
infinite residue field. Let $I$ be an $\fm$-primary ideal of $R$ and $J$ be a
minimal reduction of $I$. In this paper we show that if $\widetilde{I^k}=I^k$
and $J\cap I^n=JI^{n-1}$ for all $n\geq k+2$, then $\widetilde{I^n}=I^n$ for
all $n\geq k$. As a consequence, we can deduce that if $r_J(I)=2$, then
$\widetilde{I}=I$ if and only if $\widetilde{I^n}=I^n$ for all $n\geq 1$.
Moreover, we recover some main results [\ref{Cpv}] and [\ref{G}]. Finally, we
give a counter example for question 3 of [\ref{P1}].
