Non-Noetherian rings with an ideal not containing a product of prime ideals

Let the ring $R=\{ (a_n)_{n \in \mathbb{N}} \in \mathbb{Z}^{\mathbb{N}}\mid a_{n+1}=a_n\text{ for } n\text{ sufficiently large}\}$ and the ideal $I=(0)$.

For all $i\in \mathbb{N}$, let $e_i=(a_{i,n})_{n \in \mathbb{N}}$ with $a_{i,n}=1$ if $i=n$ and $a_{i,n}=0$ if $i \neq n$.

Let $P$ be a prime ideal.

If $i \neq j$, $e_i e_j=0$, so $e_i \in P$ or $e_j \in P$.

If there exists $i \in \mathbb{N}$ such that $e_i \notin P$, we have $e_j \in P$ for all $j \neq i$.

So $\bigoplus_{j \neq i}\mathbb{Z}e_j\subset P.$

If we choose a finite number of prime ideals $P_1,...,P_k$ with $\bigoplus_{j \neq i_m} \mathbb{Z} e_j \subset P_m$ for $m=1,...,k$,

we have $I=(0) \neq \bigoplus_{j \neq i_1,...,i_k} \mathbb{Z} e_j \subset P_1P_2\cdots P_k$.

So $I$ doesn't contain a product of prime ideals.