# Irreducible Elements, Units, UFD

Let $P$ be a set of positive prime numbers. Let $\mathbb{Z}_{P}$ be the collection of all rational numbers of the form $a/b$, where $a,b$ are integers, $b$ not in $0$, and for all $p \in P$, $p$ does not divide $b$. I need to figure out the units and the irreducible elements of $\mathbb{Z}_{P}$. Also need to find how many association classes of irreducible elements are there in $\mathbb{Q_{\{2,3\}}}$ and show that $\mathbb{Z}_{P}$ is a unique factorization domain.

So far I have that $a/b$ is a unit if $a$ is not $0$ and $a$ is not divisible by any $p \in P$. Therefore, $b/a \in \mathbb{Z}_{P}$ and so $(a/b)*(b/a) = 1$.

I am not sure how to get the irreducible elements. I know that if $b|a$ then $b$ is a unit or $b$ is equivalent to $a$ and $a$ is not a unit.

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Yes my apologies! @DonAntonio –  kkkk Aug 15 '13 at 11:45
First, I think you shouldn't call your ring $\Bbb{Q}_p$; this is reserved for the field of p-adic numbers, which is not what you mean in your question. Maybe you should call it $\Bbb{Z}_P$, because it really is some sort of localization of the ring $\Bbb{Z}$. Second, your set $P$ must consist of prime numbers, not just positive integers, for otherwise you won't end up with a ring. For example, take $P=\{4\}$ and consider the product $\frac{1}{2} \cdot \frac{1}{2}=\frac{1}{4}$. This shows you that $\Bbb{Z}_P$ is not necessarily closed under multiplication. –  Nils Matthes Aug 15 '13 at 12:16