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Let $R$ a commutative ring with one and $I, J \triangleleft R$ two ideals. The saturated ideal $I^{sat}_J$ with respect $J$ is the ideal

$$(I : J^\infty )= \cup_{n \geq 1} (I:J^n)$$

where $(I:J^n)= \{r \in R \mid rJ^n \subset I \}$. Fine.

Which intuition could one have thinking about this construction and why is it so fruitful? Is there any way to understand what happens with an ideal after saturation? On the geometrical part it is know that

$V(((I : J^\infty ))= \overline{V(I) \backslash V(J)} \subset \operatorname{Spec} R$. the bar is the closure wrt Zariski topology on $\operatorname{Spec} R$. Is this the only geometric way one should think about them?

if we assume that $I$ is saturated wrt $J$, ie $(I : J^\infty )=I$ this leads to odd implication $V(I)= \overline{V(I) \backslash V(J)}$. What does this mean, how these guys look like? Is there any "picture" one should have in mind? How they behave wrt localizations and taking radicals?

Another as the title suggests more important facet of my question is what advantages have the saturated ideals in contrast to non saturated from viewpoint of computational algebra, ie when one deals with concrete computations of eg radicals or minimal generator systems of ideals in quotient rings $k[x_1,x_2,...,x_n]/I$? Do saturated ideals have from this point of view nice features?

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    $\begingroup$ For example in $k[x,y]$ with $I=(x^m,y)$ and $J=(x)$, do you know what are $(I:J^n)$ and $(I:J^\infty)$? I apologize if this is too basic, but just in case you haven’t done examples like these, they might help. $\endgroup$
    – Zach Teitler
    Commented Jun 16, 2020 at 1:12
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    $\begingroup$ @ZachTeitler: if I am not wrong then $(I:J^\infty)$ and $(I:J^n)$ for $n \ge m$ is $(y)$ and $(I:J^n)$ for $n < m$ is $(x^{m-n},y)$. Geometrically this reminds me also of strict transforms in the theory of blow ups where $J$ is the ideal defining the closed locus where the blow up isn't an isomorphism. $\endgroup$
    – user267839
    Commented Jun 16, 2020 at 2:12
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    $\begingroup$ do you know if and why the saturated ideals are interesting for people working with computer algebra systems (eg Macaulay2) mainly interested on fast implementations of algorithms useful for computation of interesting ring/ ideal properties? $\endgroup$
    – user267839
    Commented Jun 16, 2020 at 5:25
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    $\begingroup$ One situation is to solve a polynomial system in affine space. You can complete to projective space, but there might be unwanted solutions at infinity. So remove them by saturating wrt the hyperplane at infinity. More generally instead of projective space, maybe a different completion: eg, maybe your affine space is a product (the polynomials are bihomogeneous or whatever), so you decide to complete to a product of projective spaces. Or your original system was on some other affine variety, not necessarily affine space. $\endgroup$
    – Zach Teitler
    Commented Jun 23, 2020 at 18:21
  • $\begingroup$ @user267839 No, the satuation is the entire ring $k[x,y]$ since $1\cdot x^m\in (x^m,y)$. $\endgroup$
    – AG learner
    Commented Aug 24 at 23:24

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