Let $t \in \mathbb{N}$, let $p_1, ...,p_t$ be $t$ distinct prime numbers. Let $R$ = { $\alpha \in \mathbb{Q}$ : $\alpha=\frac{m}{n}$ for some $m \in \mathbb{Z}$ and $n \in \mathbb{N}$ such that $n$ is divisible by none of $p_i$}. Show that $R$ is a subring of $\mathbb{Q}$ which has exactly $t$ maximal ideals.
I'm getting stuck here because i have no idea to solve this problem. Checking $R$ is a subring of $\mathbb{Q}$ is trivial but i don't know how to solve the next part. I wondering that whether every $p_i$ is belong to some maximal ideal $P_i$ and every $P_j$ are distinct.
I need help.
Many thanks.