Let us call an ideal $I$ of a Noetherian local ring $(R, \mathfrak m)$ special if $\mathfrak m I \ne \mathfrak m ( I : \mathfrak m)$ .
I have the following questions:
(1) Let $I$ be a special ideal in a Noetherian local ring $(R, \mathfrak m)$, then is it true that $I_P (=IR_P)$ is special in $(R_P, PR_P)$ for every prime ideal $P$ of $R$ containing $I$ i.e. for every prime ideal $P \in Ass (R/I)$ ?
(2) Let $I$ be an ideal in a Noetherian ring $R$ such that $I_{\mathfrak m}$ is special in $ R_{\mathfrak m}$ for every maximal ideal $\mathfrak m \in Ass (R/I)$ , then is it true that $I_P (=IR_P)$ is special in $R_P$ for every prime ideal $P \in Ass (R/I) $ ?
If either of (1) or (2) is not true in general, I would like to know if they are true for some special kind of rings (like say Cohen-Macaulay rings ...)
Note that easily (2) implies (1) .
Some remarks about special ideals : $I$ is special in $(R, \mathfrak m)$ if and only if $(I: \mathfrak m)\ne (\mathfrak m I :\mathfrak m)$ if and only if $( (I:\mathfrak m)/I ). (\mathfrak m/I \mathfrak m) \ne 0$ if and only if $I \hat R$ is special in $\hat R$ ($ \hat R $ being the completion of $R$ w.r.t. the maximal ideal). Also note that if $I$ is special, then depth $R/I=0$ .