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rings of rational numbers
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(Theorem)
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The criterion for a non-empty subset  of a given ring  for being a subring of  , is that  contains always along with its two elements also their difference and product. Since the field
 of the rational numbers is (isomorphic to) the total ring of quotients of the ring
 of the integers, any rational number is a quotient
 of two integers  and  . If now  is an arbitrary subring of
 and
with
 (and
 ), then one must have
Therefore, the set of possible denominators of the elements of  is closed under multiplication, i.e. it forms a multiplicative set. We can of course confine us to subsets  containing only positive integers. But along with any positive integer  , the set  has to contain also all positive divisors, inclusive 1 and the prime divisors of the number  , since the factorisation  of the denominator of an element
 of  implies that the multiple
 belongs to  . Accordingly,  consists of 1, a certain set of positive prime numbers and all finite products of these, thus being a free monoid on the set of those prime numbers.
Since  contains all multiples of each of its elements, it is apparent that the set of possible numerators form an ideal of
 .
 Theorem. If  is a subring of
 , then there are a principal ideal  of
 and a multiplicative subset  of
 such that  is a free monoid on certain set of prime numbers and any element
 of  is characterised by
The positive generator  of  does not belong to  except when it is 1.
Note. Since  may be greater than 1, the ring  is not necessarily the ring of quotients
 , e.g. in the case
Examples.
1. The ring
 of the p-integral rational numbers where
 . E.g. the 2-integral rational numbers consist of fractions with arbitrary integer numerators and odd denominators, for example
 .
2. The ring
 of the decimal fractions where
 .
3. The ring of the terminating binary or dyadic fractions with any integer numerators but denominators from the set
 .
4. If  , the subring of
 is simply some ideal  of the ring
 .
All the subrings of
 (except  ) have
 as their total ring of quotients.
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"rings of rational numbers" is owned by pahio.
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See Also: localization, therefore sign
| Other names: |
subrings of rationals, subrings of  |
| Also defines: |
dyadic fraction, p-integral rational numbers,  -integral rational number |
This object's parent.
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Cross-references: decimal fractions, odd, fractions, ring of quotients, generator, multiplicative subset, principal ideal, ideal, numerators, free monoid, finite, prime numbers, implies, number, positive, multiplicative set, multiplication, closed under, denominators, quotient, rational number, integers, total ring of quotients, isomorphic, rational numbers, field, product, difference, contains, subring, ring, subset
There are 4 references to this entry.
This is version 13 of rings of rational numbers, born on 2008-05-18, modified 2008-05-25.
Object id is 10600, canonical name is RingsOfRationalNumbers.
Accessed 643 times total.
Classification:
| AMS MSC: | 11A99 (Number theory :: Elementary number theory :: Miscellaneous) | | | 13B30 (Commutative rings and algebras :: Ring extensions and related topics :: Quotients and localization) |
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Pending Errata and Addenda
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