Let $R$ be the local ring of a complex curve singularity. (Can assume the singularity planar, the ring locally analytic or formal.) Let $\bar{R}$ be the normalization, let $R\subset R'\subset \bar{R}$ be an intermediate extension, corresponding to the factorization of the normalization map: $Spec(\bar{R})\to Spec{R'}\to Spec{R}$.

The relative conductor: $I^{cd}_{R'/R}:=(r\in R: r R' \subset R)$ is an ideal in $R$ (and in $R'$).

The conductor for the normalization $I^{cd}_{\bar{R}/R}$ is well studied. Any reference for the relative conductor $I^{cd}_{R'/R}$?

Specific questions:

The relative conductor can be defined as $I^{cd}_{R'/R}=R:R'$. (Just another way to write the same thing.) Can we also say: $R'=R:I^{cd}_{R'/R}$? (i.e. $R:(R:R')=R'$). In words: for a given conductor $I$ take the maximal extension, whose conductor is $I$. Will this reproduce the initial extension?

For a general ideal $I\subset R$, not a conductor of some extension, define $R'=R:I\subset\bar{R}$. (i.e. the maximal subring of the integral closure such that $R'I\subset R$). Then $I^{cd}_{R'/R}$ contains $I$, but in general is bigger. Any conditions on $I$ to ensure: $R:(R:I)=I$?

Can this be somehow generalized to higher dimensions? At least, is the following always true: $R:(R:(R:I))=R:I$ ?

Any reference?

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