To prove a given set is an UFD

Asked Dec 13 '21 at 12:58

Active Dec 13 '21 at 12:58

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The problem is to show that the set $R$ is an UFD, where

$R := \{a\in \mathbb{Q}| a=\frac{n}{2^k} \text{ for some }n, k\in \mathbb{Z},k\geq0 \}$

I am thinking of something like $\mathbb{Q}[x]/(2x-1)$ but I cannot proceed from here.

Should I go through the definition of an UFD, that are

(1) unique factorization

(2) the Ascending Chain Condition?

If not, in general how could I show that a given set is an UFD? Since for other domains like ED and PID, the ideas are quite straight forward. Any help would be appreciated.

asked Dec 13 '21 at 12:58

mibamiba

1

It's a special case of UFDs are preserved under localization, where $S$ (the denominator monoid) is that generated by $2$. Or, directly, you could extend (tweak) the proofs from $\Bbb Z$ after showing that the primes in $R$ are all the odd primes in $\Bbb Z$. – Bill Dubuque Dec 13 '21 at 13:09
"how could I show that a given set is UFD?". See this post, for example. It is of course again related to localisation. – Dietrich Burde Dec 13 '21 at 17:55

To prove a given set is an UFD

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