rings of rational numbers

The criterion for a non-empty subset R of a given ring Q for being a subring of Q, is that R contains always along with its two elements also their difference and product.  Since the field of the rational numbers is (isomorphicPlanetmathPlanetmathPlanetmath to) the total ring of quotients of the ring of the integers, any rational number is a quotient mn of two integers m and n.  If now R is an arbitrary subring of and


with  m1,n1,m2,n2  (and  n1n20), then one must have


Therefore, the set of possible denominators of the elements of R is closed under multiplication, i.e. it forms a multiplicative set.  We can of course confine us to subsets S containing only positive integers.  But along with any positive integer n0, the set S has to contain also all positive divisorsMathworldPlanetmathPlanetmathPlanetmath (http://planetmath.org/Divisibility), inclusive 1 and the prime divisorsPlanetmathPlanetmathPlanetmath (http://planetmath.org/FundamentalTheoremOfArithmetics) of the number n0, since the factorisation  n0=uv  of the denominator of an element mn0 of R implies that the multipleMathworldPlanetmath (http://planetmath.org/GeneralAssociativity)  umuv=mv  belongs to R.  Accordingly, S consists of 1, a certain set of positive prime numbersMathworldPlanetmath and all finite products of these, thus being a free monoid on the set of those prime numbers.

Since R contains all of each of its elements, it is apparent that the set of possible numerators form an ideal of .

Theorem.  If R is a subring of , then there are a principal idealMathworldPlanetmathPlanetmath (k) of and a multiplicative subset S of such that S is a free monoid on certain set of prime numbers and any element mn of R is characterised by


The positive generatorPlanetmathPlanetmathPlanetmath k of (k) does not belong to S except when it is 1.

Note.  Since k may be greater than 1, the ring R is not necessarily the ring of quotients S-1, e.g. in the case



1.  The ring  R:=S-1  of the p-integral rational numbers (http://planetmath.org/PAdicValuation) where
S={thepowerproductsofallpositiveprimesexceptp}.  E.g. the 2-integral rational numbers consist of fractions with arbitrary integer numerators and odd denominators, for example 10001001.

2.  The ring  R:=S-1  of the decimal fractions  where  S={thepowerproductsof 2and 5}.

3.  The ring of the or dyadic fractions with any integer numerators but denominators from the set  S={1, 2, 4, 8,}.

4.  If  S={1},  the subring of is simply some ideal (k) of the ring .

All the subrings of (except the trivial ring {0}) have as their total ring of quotients.

Title rings of rational numbers
Canonical name RingsOfRationalNumbers
Date of creation 2014-03-18 15:35:06
Last modified on 2014-03-18 15:35:06
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 17
Author pahio (2872)
Entry type Theorem
Classification msc 13B30
Classification msc 11A99
Synonym subrings of rationals
Synonym subrings of
Related topic LocalizationMathworldPlanetmath
Related topic ThereforeSign
Defines dyadic fraction
Defines p-integral rational numbers
Defines p-integral rational number