I was wondering whether there are is a nice characterization of commutative rings $R$ where for ideals $I,J,K$ we always have: $$(I+K)\cap (J+K) = I\cap J +K $$
Motivation:
There is an extended version of the classical chinese remainder theorem in number theory which goes as follows:
Generalized Chinese Remainder theorem. If $a_1,a_2,...,a_n$ and $m_1,m_2,..,m_n$ are integers such that $a_i \equiv a_j \,\text{mod} \gcd(a_i,a_j)$ for all $i,j\in \{1,\dots,n\}$ then the system of equations $$ \begin{cases} x \equiv a_1 \mod m_1 \\ x \equiv a_2 \mod m_2 \\ \dots \\ x \equiv a_n \mod m_n \\ \end{cases}$$ Has a solution which is unique $\text{mod}\, \text{lcm}(a_1,\dots,a_n)$
The usual CRT has a general version in a commutative ring setting, and we can do the same for the generalized CRT:
Generalized CRT: Commutative Rings. Let $R$ be a commutative ring (with identity) and $I_1,\dots,I_n$ ideals of $R$. Then the canonical map $$\varphi : R \rightarrow R/I_1\times R/I_2\times \cdots R/I_n $$ has kernel $I_1\cap\dots\cap I_n$ and has image precisely all the $(\overline{a_1},\dots,\overline{a_n})$ with $a_k = a_l \,\text{mod}\, (I_k+I_l)$ for all $k,l \in \{1,\dots,n\}$
The case where $n=2$ is proven quite easily in the general case but when translating the proof of $\mathbb{Z}$ for $n\geq 3$ we make use of the fact that $$(I_1+I_n)\cap(I_2+I_n)\cap\dots\cap(I_{n-1}+I_n) = I_1\cap\dots\cap I_{n-1} + I_n$$ Which is true for ideals in $\mathbb{Z}$ but not in general! In fact, I noted that this is also a necessary condition: for a commutative ring $R$ the generalized CRT holds if and only if for all ideals $I,J,K$ of $R$ we have $$(I+K)\cap (J+K) = I\cap J +K$$
So knowing how to handle this last condition might be interesting.
Thoughts so far
The proof in $\mathbb{Z}$ boils down to the fact that $$\text{lcm}(\gcd(a,c),\gcd(b,c)) = \gcd(\text{lcm}(a,b),c)$$ for all integers $a,b,c$ which can be checked by looking at the exponents of the prime powers. Hence, since every PID is a UFD the equality holds for every PID with the same proof.
However, I know from this question that there exists counterexamples, even in the UFD $\mathbb{Z}[X]$ which is already quite disturbing. Does the equality hold in Dedekind rings, local rings?
Thanks in advance!
