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Let $R$ be a commutative Noetherian ring. Then, I cannot construct a counterexample (or proof) for the question posed in the title.

If $R$ is a polynomial ring and all ideals involved are monomial ideals, then this statement is true since it will follow that the minimal generators of $I$, $J$, and $K$ can be contained in polynomial rings in disjoint variables. More generally, it is clear that $K \subset (IJK : I \cap J) \subset (IJK : IJ)$. At this point, it is not clear to me how to proceed.

Any help is appreciated.

Rellek
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    Find a ring with $IJ\neq I\cap J$, then take $K=(0)$. – Arturo Magidin Jun 23 '21 at 20:19
  • Ok, fine. Let us assume then that none of the ideals contain each other. – Rellek Jun 23 '21 at 20:58
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    Please put all your assumptions in the post; don't force people to guess them, or to dig through comments to find them. – Arturo Magidin Jun 23 '21 at 21:25
  • What sort of counterexamples have you looked for so far? For example, you've considered monomial ideals of polynomial rings, but what about monomial ideals of the quotient of a polynomial ring by a monomial ideal? (e.g. rings like $k[x, y]/(x^2, xy)$) – Daniel Hast Jun 23 '21 at 23:13

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