I am looking for a commutative ring $R$ with identity which has a family $F:=\{p_i\}_{i\in A}$ of prime ideals such that for each $n\in \mathbb{N}$ there exists $m\in \mathbb{N}$ with $n\leqslant m$ and a partition $\{\{p_i\}_{i\in A_1},...,\{p_i\}_{i\in A_m}\}$ of $F$ with the following property:
1) For each $j\in \{1,...,m\}$, if $\cap_{i\in A'_j}p_i +\cap_{i\in A''_j}p_i=R$, where $A_j=A'_j\cup A''_j$, then there exists an idempotent element $e^2=e\in R$ such that $e\in \cap_{i\in A'_j}p_i $ and $1-e\in \cap_{i\in A''_j}p_i$.
2) The ideals $\cap_{i\in A_1}p_i, \cap_{i\in A_2}p_i,..., \cap_{i\in A_m}p_i$ are pairwise comaximal.
3) For $k\not= j $ there is no idempotent element $f^2=f\in R$ such that $f\in \cap_{i\in A_k}p_i $ and $1-f\in \cap_{i\in A_j}p_i $.
