rings of rational numbers
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 (http://planetmath.org/Divisibility), inclusive 1 and the prime divisors (http://planetmath.org/FundamentalTheoremOfArithmetics) of the number , since the factorisation of the denominator of an element of implies that the multiple (http://planetmath.org/GeneralAssociativity) 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 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
1. The ring of the p-integral rational numbers (http://planetmath.org/PAdicValuation) 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 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 the trivial ring ) have as their total ring of quotients.
|Title||rings of rational numbers|
|Date of creation||2014-03-18 15:35:06|
|Last modified on||2014-03-18 15:35:06|
|Last modified by||pahio (2872)|
|Synonym||subrings of rationals|
|Defines||p-integral rational numbers|
|Defines||-integral rational number|