On the computation of the Ratliff-Rush closure, associated graded ring and invariance of a length

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