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Let $A$ be a commutative ring and $B = S^{-1}A$ be its localisation 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.)

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