rings of rational numbers

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

ℚ

of the

rational numbers

is (

isomorphic

to) the

total ring of quotients

of the ring

ℤ

of the

integers

, any rational number is a

quotient

m

n

of two integers

m

and

n

.  If now

R

is an arbitrary subring of

ℚ

and

m

1

n

1

,

m

2

n

2

∈

R

with

m

1

,

n

1

,

m

2

,

n

2

∈

ℤ

(and

n

1

⁢

n

2

≠

0

), then one must have

m

1

⁢

n

2

-

m

2

⁢

n

1

n

1

⁢

n

2

∈

R

,

m

1

⁢

m

2

n

1

⁢

n

2

∈

R

.

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

n

0

, the set

S

has to contain also all

positive

divisors

(

http://planetmath.org/Divisibility

), inclusive 1 and the

prime divisors

(

http://planetmath.org/FundamentalTheoremOfArithmetics

) of the number

n

0

, since the factorisation

n

0

=

u

⁢

v

of the denominator of an element

m

n

0

of

R

implies that the

multiple

(

http://planetmath.org/GeneralAssociativity

)

u

⋅

m

u

⁢

v

=

m

v

belongs to

R

.  Accordingly,

S

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

R

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

ℤ

.

∴

Theorem.

If

R

is a subring of

ℚ

, then there are a

principal ideal

(

k

)

of

ℤ

and a multiplicative subset

S

of

ℤ

such that

S

is a free monoid on certain set of prime numbers and any element

m

n

of

R

is characterised by

{

m

∈

(

k

)

,

n

∈

S

.

The positive

generator

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

⁢

ℤ

, e.g. in the case

R

=

{

2

⁢

a

3

s

⋮

a

∈

ℤ

,

s

∈

ℤ

+

}

.

Examples.

1.  The ring

R

:=

S

-

1

⁢

ℤ

of the

p-integral rational numbers (

http://planetmath.org/PAdicValuation

)

where

S

=

{

the

⁢

power

⁢

products

⁢

of

⁢

all

⁢

positive

⁢

primes

⁢

except

⁢

p

}

.  E.g. the 2-integral rational numbers consist of fractions with arbitrary integer numerators and odd denominators, for example

1000

1001

.

2.  The ring

R

:=

S

-

1

⁢

ℤ

of the

decimal fractions

where

S

=

{

the

⁢

power

⁢

products

⁢

of

⁢

2

⁢

and

⁢

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

ℚ

is simply some ideal

(

k

)

of the ring

ℤ

.

All the subrings of

ℚ

(except the

trivial ring

{

0

}

) have

ℚ

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

ℚ

Related topic

Localization

Related topic

ThereforeSign

Defines

dyadic fraction

Defines

p-integral rational numbers

Defines

p

-

integral

rational number

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