How to classify the rings of fractions of a principal ideal domain?

Question

Let $A$ be a principal ideal domain and let $K$ be its field of fractions. I proved
a) Every ring $B$ such that $A \subset B \subset K$ is a ring of fractions of $A$,
and
c) Show that any ring of fractions of $A$ is principal.

But I'm having trouble with

b) Classify the rings of fractions of $A$.

The hint given is to consider the prime elements of $A$ which are invertible in $S^{-1}A$. However, it still isn't clear what sort of classification the author expects, and how to use the hint. Help would be much appreciated.

Bill Dubuque · Accepted Answer · 2015-04-19T21:25:22.513

4

Hint $\ $ The overring is uniquely determined by the set of elements inverted, which is uniquely determined by the set of primes inverted. This yields a bijection between such overrings and subsets of primes of $A$.

In further detail, the set $\,S\subset A\,$ of elements that become units (invertibles) in the overring is a saturated monoid $\,S,\,$ i.e. $\,ab\in S\iff a,b\in S.\,$ One easily checks that such monoids are characterized (generated) by their prime elements, using the fact that $A$ is a PID so a UFD, so every element is $\,S\,$ has a (unique) prime factorization.

edited Apr 19 '15 at 21:25

answered Apr 19 '15 at 21:10

Bill Dubuque

272,048

Thanks. Can I say something more about subsets of primes of A? – odnerpmocon Apr 19 '15 at 21:12
@odnerpmocon Not in general. – Bill Dubuque Apr 19 '15 at 21:14
1

So your hint pretty much says everything that can be said here? – odnerpmocon Apr 19 '15 at 21:15
@odnerpmocon Yes. You might find it helpful to consider the case $,A = \Bbb Z\ \ $ – Bill Dubuque Apr 19 '15 at 21:17
@odnerpmocon I added a further remark. – Bill Dubuque Apr 19 '15 at 21:25
I saw it, thanks. – odnerpmocon Apr 19 '15 at 21:34

How to classify the rings of fractions of a principal ideal domain?

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