Distributive property of sum and intersection of ideals in a polynomial ring

Question

Let $S=k[x_1,\dots,x_n]$, where $k$ is a field, and $I,J,K$ be three ideals of $S$. Are the following right or are they wrong?

1. $ I+ (J\cap K)=(I+J)\cap (I+ K)$
2. $ I\cap (J+ K)=(I\cap J)+ (I\cap K)$

Answer 1

The rings in which 1 holds for all $I,J,K$ - or equivalently, those in which 2 holds for all $I,J,K$ - are called arithmetical. In a non-arithmetical ring, 1 and 2 would hold only in the special case where $I$ contains at least one of $J$ or $K$: see e.g. When does the distributive law apply to ideals in a commutative ring? and Distributivity of intersection over addition in ideals.

A domain (i.e. a ring where products of non-zero elements are non-zero) that happens to be an arithmetical ring is called a Prüfer domain. As the answers linked above also explain, there are a number of natural conditions under which a domain will be Prüfer - for example, $D$ is Prüfer if and only if Gauss's lemma holds in $D$; or $D$ is Prüfer if and only if the Chinese remainder theorem holds in $D$. Thus 1 and 2 do hold in general in e.g. any principal ideal domain: see Distributive law of ideals in $\mathbf Z$ relating $\cap$ & $+$.

But by contrast, it turns out that multivariate polynomial domains are never Prüfer! See When do (multivariate) polynomial rings fail to be Prüfer rings?. So the answer to your particular problem is that 1 and 2 don't hold except (as mentioned above) if $I \supset J$ or $I \supset K$.