Distributive law of ideals in $\mathbf Z$ relating $\cap$ & $+$ Let $\mathfrak a, \mathfrak b$ and $\mathfrak c$ be ideals in $\mathbf Z$.
Then show that
$$ \mathfrak a \cap (\mathfrak b + \mathfrak c) = \mathfrak a \cap \mathfrak b + \mathfrak a \cap \mathfrak c .$$
It is easy to prove $\supset$ side but $\subset$ side is difficult for me. 
My questions are following.
1. How to prove $\subset$ side.
2. Is it true for all of principal ideal domains? 
 A: @CPM Thank you for your link.
I just have obtained an elementary proof.  

In a principal ideal domain,  the relation
  $(a)\cap ((b)+(c)) = (a)\cap (b) + (a)\cap (c) $ holds.

I'll show the $\subset$ side.
Let $(b)+(c) = (d)$ and $(a)\cap (d) = (e)$
and we can write $b=b'd, c=c'd$ with $sb'+tc'=1$  and $ e=fa = gd$
Then
$$ e = e \cdot 1 = e(sb'+tc') = esb' + etc' $$
Since $ e=fa=gd ,  esb'\in (a)\cap (b)$ and $ etc' \in (a)\cap (c)$
and the proof is completed.
A: A ring which satisfies this property is called arithmetical.  This paper has some nice results related to this: https://www.math.purdue.edu/~heinzer/preprints/irr15.pdf
For domains this is the same as being Prufer.  I would direct you here, there is a very informative answer here: When does the modular law apply to ideals in a commutative ring
Edit: I got so excited about the second question, I suppose I forgot to comment on the first one.  I think it is easiest for the integers to think about 
$$(a) + (b)= (gcd(a,b)) \ \text{ and } \ (a)\cap (b)=(lcm(a,b)).$$  Then you have reduced the question to needing to show that: 
$$lcm(a, gcd(b,c))=gcd(lcm(a,b),lcm(a,c))$$
This is done here: https://proofwiki.org/wiki/GCD_and_LCM_Distribute_Over_Each_Other
Since you have unique factorization in the integers, it is pretty clear to see by just looking at the prime factorizations why this is true.  
