# Does the contraction from the localized ring preserve colon ideals and ideal sums/products?

Let $A$ be a commutative ring and $B = S^{-1}A$ be its localization with respect to a certain multiplicative subset of $A$.

Consider the contraction (in $A$) of colon ideals and ideal sums and ideal products (in $B$) as long as they make sense.

Do contracted ideals still possess the original characteristics?

That is, will the contraction of colon ideals (resp. of sums, resp. of products) in $B$ be colon ideals (resp. sums, resp. products) of the corresponding contracted ideals in $A$?

I suspect there are counterexamples if $A$ is not noetherian, but I have no idea how to tackle this.

(Thanks for pointing out obscurity. I hope this time it is more legible.)

• In any case, for anyone out there who might possibly be able to help (thanks!), my edit read: "Does anyone has an answer to this, especially for the case where $A$ is an integral domain? Also, is $A$ being Noetherian a sufficient condition for ideal sums/products to be preserved?"
– Ligo
Commented Mar 1, 2016 at 10:24
• – Ligo
Commented Mar 1, 2016 at 11:38

For sums:

No. Let $R=K[X,Y]$, $f=X+Y$, $S=\{1,f,f^2,\dots\}$, $I=(X)$, and $J=(Y)$. Then $S^{-1}I\cap R=I$ and $S^{-1}J\cap R=J$ (since $I,J$ are prime ideals and $I\cap S=J\cap S=\emptyset$), while $(S^{-1}I+S^{-1}J)\cap R=S^{-1}(I+J)\cap R=S^{-1}R\cap R=R$ (since $(I+J)\cap S\ne\emptyset$).

For products:

No. Let $R=K[X,Y,Z]/(XY-Z^2)$, $S=\{1,y,y^2,\dots\}$, and $I=J=(x,z)$. Then $S^{-1}I\cap R=I$ (since $I$ is a prime ideal and $I\cap S=\emptyset$), while $(S^{-1}IS^{-1}J)\cap R=S^{-1}I^2\cap R\supsetneq I^2$ since $x=\frac{xy}{y}=\frac{z^2}{y}\in S^{-1}I^2\cap R$ and $x\notin I^2$.

For colons:

Yes.

$$(S^{-1}I:S^{-1}J)\cap R=(S^{-1}I\cap R):(S^{-1}J\cap R).$$

• We also have $S^{-1}(I:J)\cap R\subseteq(S^{-1}I:S^{-1}J)\cap R$ with equality for $J$ finitely generated. There are examples where this inclusion is strict. Commented Mar 12, 2016 at 11:59
• Thank you very much for the answer. Especially, the counter examples are wonderful. Commented Mar 13, 2016 at 11:18
• Can anyone give geometric intuition for how these examples were constructed? Commented Apr 8, 2022 at 14:13