Let $D$ be a principal ideal domain. Let $I,J,K$ be ideals of D. Show that $I+(J\cap K) = (I+J)\cap(I+K)$

Question

I'm trying to show that in a principal ideal domain $D$, it holds $I+(J\cap K) = (I+J)\cap(I+K)$, where $I,J,K$ are ideals of $D$.

My attempt is the following:

1 - $I+(J\cap K) \subset (I+J)\cap(I+K)$

Since $D$ is a principal ideal domain, there exist $a_1,a_2,a_3\in D$ such that $I=a_1 D$, $J=a_2D$ and $K=a_{3}D$. Additionally, note that $J\cap K$ is an ideal of $D$. Therefore, there exists $a_4\in D$ such that $(J\cap K) =a_4 D$. Using this, we have that $dD = a_1 D+a_4 D$ for some $d\in D$. Now, noting that $a_4D\subset J$ and $a_4 D \subset K$, it follows that $dD = a_1 D+a_4 D\subset I + J$ and $dD = a_1 D+a_4 D\subset I + K$, which means that the inclusion $I+(J\cap K) \subset (I+J)\cap(I+K)$ holds.

I'm not sure if this first part is complete.

Additionally, I do not know how to approach the second inclusion $(I+J)\cap(I+K)\subset I+(J\cap K)$.

Any help? Thanks

See art of problem solving. It holds for Prüfer domains, so for PIDs. — Dietrich Burde, Jul 22 '24 at 19:22
Notice that $aD + bD = \textrm{gcd}(a,b)D$ and$aD \cap bD = \textrm{lcm}(a,b)D$ . — eti902, Jul 22 '24 at 19:22
Related answer at: Distributive law of ideals in $\mathbf Z$ relating $\cap$ and $+$. — K B Dave, Jul 22 '24 at 19:28
In a PID this is equivalent to gcd distributes over lcm, which is proved here in many places. — Bill Dubuque, Jul 22 '24 at 20:30
This is not a duplicate of https://math.stackexchange.com/q/1311338/491450 ! The question here is whether $I + (J \cap K) = (I + J) \cap (I + K)$. The question there is whether $I \cap (J + K) = I \cap J + I \cap K$. — Smiley1000, Nov 05 '24 at 14:35
This is actually a duplicate of https://math.stackexchange.com/q/119833/491450 . — Smiley1000, Nov 05 '24 at 14:37

Let $D$ be a principal ideal domain. Let $I,J,K$ be ideals of D. Show that $I+(J\cap K) = (I+J)\cap(I+K)$

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