By an arithmetical ring is understood a commutative ring $R$ with identity for which the ideals form a distributive lattice, i.e., for which $(I+J)\cap K=(I\cap K)+(J\cap K)$ for all ideals $I, J$ and $K$ of $R$. Also, a ring $R$ is called a completely arithmetical ring if for every ideal $I$ and every nonempty family of ideals $\{J_i\}_{i\in A}$ one has $I +\bigcap_{i\in A}J_i=\bigcap_{i\in A} (I+J_i)$. I am looking for some references for studying rings with the following $X$-property, which is not equivalent to the completely arithmetical property see [1]:
A ring $R$ has $X$-property whenever for every ideal $I$ and every nonempty family of ideals $\{J_i\}_{i\in A}$ one has $I \cap(\sum_{i\in A}J_i)=\sum_{i\in A} (I\cap J_i)$.
[1].https://www.tandfonline.com/doi/abs/10.1080/00927872.2013.804924
