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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.

Liêu Long Hồ
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  • Hint: show that $R$ is a localization of $\mathbb{Z}$ at some multiplicative subset. What do you know about ideals of localizations? – Aphelli Nov 04 '20 at 15:32
  • Here's a slightly different hint that, to my taste, is easier to think about: a field $k$ has a unique maximal ideal because any $\alpha \in k^*$ generates the unit ideal: you can write $1 = a_1x_1 + \cdots +a_n x_n$ with $n=1$, $x_1 = \alpha$, and $a_1 = \alpha^{-1}$. For which elements of $R$ does this property fail? – Tabes Bridges Nov 04 '20 at 17:52

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