## Rings, Integral Domains and Fields## Philip J. Erdelsky## March 24, 2007 |

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A __ring__ is a set *R* and two binary operations, called addition
and multiplication, with the following properties:

- The ring is a commutative group under addition.
- Multiplication is associative:
*a(bc) = (ab)c* - Multiplication distributes over addition:
*a(b+c) = ab + ac*

(a+b)c = ac + bc

The properties of multiplication involving zero (the additive identity) and signed ring elements are the same as those derived for the integers (which are a ring), and the proofs are the same, but slightly more complicated because multiplication is not necessarily commutative:

*0x = x0 = 0**(-x)y = x(-y) = -(xy)**(-x)(-y) = xy*

A __ring isomorphism__ between the rings *R* and *S*
is a one-to-one correspondence *f: R ⟶ S*
which preserves the ring operations:

*f(x+y) = f(x) + f(y)**f(xy) = f(x) f(y)*

There are minor variations in the definition of a ring; what we have
presented is the minimal definition. Some authors
require that a ring have a __unit__, which is an identity element
for multiplication; i.e.
a number *1* such that *1a = a1 = a* for every element
*a* of the ring. Also, it is often required that
*0 ≠ 1*, because a ring in which
*0 = 1* is a trivial ring with only one element.

A __commutative ring__ is a ring with commutative multiplication.

The integers are a commutative ring with a unit.
The even integers are a commutative ring without a unit.
The set *Z _{M}*, previously defined as the
integers

A __left ideal__ of a ring is a nonempty subset closed under
subtraction
and left multiplication by any ring element; i.e. if *x* and *y*
are in the ideal and *a* is any ring element, then *x-y*
and *ax* are in the ideal.
Similarly, a __right ideal__ of a ring is a nonempty subset closed under
subtraction
and right multiplication by any ring element; i.e. if *x* and *y*
are in the ideal and *a* is any ring element, then
*x-y* and *xa* are in the ideal. An __ideal__ is a set
that is both a left ideal and a right ideal. Obviously, in a commutative
ring there are no differences among the three kinds of ideals.

Although an ideal is required to be closed only under subtraction, it is
easy to show that it is also closed under addition. If *x* and *y*
are in the ideal, then *0* is in the ideal because it is equal
to *x-x*, *-y* is in the ideal because it is equal
to *0-y*, and *x+y* is in the ideal because it is equal
to *x-(-y)*.

Ring theory is a well-developed branch of mathematics, but we need only these basic concepts. We will deal mainly with rings that have additional properties.

An __integral domain__ is a commutative ring with unit
(and *0 ≠ 1*) in which there
are no zero divisors; i.e., *xy = 0* implies that *x=0* or
*y=0* (or both).

The integers are an integral domain; this is the reason for the name.
The set
*Z _{M}*, previously defined as the
integers

Since an integral domain is a group under addition, the order of
a nonzero element *a* is the smallest positive value of
*n*, if any,
such that *na = 0* (where *na = a+a+a+...+a* (*n* times)).
Every nonzero element has the same order as *1* because
*na = (n1)a = 0* only when *n1 = 0*.

The order must be prime. If it could be factored as *n = ab*,
then *1+1+...+1* (*a* times) and *1+1+...+1* (*b* times)
would be two nonzero elements whose product would be zero.

The order of any nonzero element of an integral domain is often called
the __characteristic__ of the integral domain,
especially when the integral domain is also a field.

An integral domain is a __field__ if every nonzero
element *x* has a __reciprocal__ *x ^{ -1}* such that

__Division__ in a field is defined in the usual way:

where the denominatorx / y = x y^{ -1},

From this definition and the properties of fields, we can derive the usual rules for operations on fractions:

*a/b = c/d*if, and only if,*ad = bc**a/b + c/d = (ad + bc) / (bd)**(a/b) (c/d) = (ac) / (bd)**(a/b)*^{ -1}= b/a*(-b)/a = b/(-a)a = -(a/b)**0/a = 0**a/1 = a*

A __subfield__ of a field is a subset which is a field under the same
addition and multiplication operations.

A rational number is a real number which can be expressed as
the quotient of two integers. The integers are an integral domain,
and the rational numbers are a field. This sort of relationship
applies more generally. Every integral domain
has a related field called its __field of quotients__, which
is the smallest field that contains a subset isomorphic to the
domain.

The relationship between the integers and the rational numbers shows how a field of quotients can be constructed.

Let *D* be an integral domain. We first define a relation on
*D ⨯ (D - {0})* as follows:

(a,b) ~ (c,d)ifad = bc

(Notice that this is *a/b = c/d* cleared of fractions.)
It is easy to show that this is an equivalence relationship.

We define addition
and multiplication on
*D ⨯ (D - {0})* as follows:

(a,b) + (c,d) = (ad + bc, bd)

(a,b)(c,d) = (ac, bd)

It can be shown that addition of equivalent pairs gives equivalent results. Hence the addition of two equivalence classes can be defined to be the class containing the sum of any elements in the two classes. Multiplication of equivalence classes can be defined in the same way.

It can be shown that the set of equivalence classes is a field under these
definitions of addition and multiplication, and that the classes
containing pairs of the form *(a,1)* are isomorphic to *D*.

Moreover, this field is the smallest such field; any other field that
contains a subset isomorphic to *D* also contains a subfield
isomorphic to the field of quotients as constructed. The isomorphism
is a mapping that
carries each quotient *a/b* of two elements of *D* to the
equivalence class containing *(a,b)*.

The field of rational numbers derived from the integers is often written
as *Q*.

An __ordered integral domain__ is an
integral domain with a subset of __positive__
elements with the following properties:

- The sum and product of two positive elements are positive.
- Zero is not positive.
- For every nonzero element
*a*, either*a*or*-a*, but not both, is positive.

The element *a* of an ordered integral domain is said to be
__negative__ if *-a* is positive.

Since either *a* or *-a* is positive when *a* is
nonzero, the product *aa*, which is equal to *(-a)(-a)*,
is positive in either case. In particular, the unit is positive.

This is called an order because a linear order of the integral domain
or field elements can be obtained by defining *a < b* when
*b-a* is positive.

The field of quotients of an ordered integral domain is ordered by defining as positive the quotient of any two positive elements of the integral domain. In fact, this is the only way of ordering the field in a way that is consistent with the ordering of the integral domain.

An ordered field is __archimedean__ if every number is less than some
multiple of the unit. The field of rational numbers is archimedean;
later we will see some non-archimedean fields.

The field *Q* of rational numbers is insufficient for many purposes. It
seems to be full of holes. For example, there is no rational number
whose square is exactly *2*, which can be shown by assuming, for
purpose of contradiction, that *m* and *n* are two integers
such that *(m/n) ^{ 2} = 2*. This would imply that

There are a number of ways to fill in the holes. One of them involves sequences and limits, which belong to the realm of analysis rather than algebra.

A sequence *{x _{1}, x_{2}, x_{3}, ...}*
of rational numbers is a

A Cauchy sequence of rational numbers does not always have a rational limit.
For example, it is fairly easy to construct a Cauchy sequence of rational
numbers that approaches the square root of *2*.

Two Cauchy sequences *{x _{1}, x_{2}, x_{3}, ...}*
and

The equivalence classes are the real numbers.

We define addition and multiplication of Cauchy sequences with term-by-term addition and multiplication:

*{x*._{1}, x_{2}, x_{3}, ...} + {y_{1}, y_{2}, y_{3}, ...} = {x_{1}+ y_{1}, x_{2}+ y_{2}, x_{3}+ y_{3}, ...}*{x*._{1}, x_{2}, x_{3}, ...} {y_{1}, y_{2}, y_{3}, ...} = {x_{1}y_{1}, x_{2}y_{2}, x_{3}y_{3}, ...}

It is easy to prove that equivalent sequences have equivalent sums, and only slightly more difficult to prove that equivalent sequences have equivalent products. Hence addition and multiplication of equivalence classes is well-defined.

The result has all the required field properties, and sequences which
have rational limits are isomorphic to *Q*.

The result is also an ordered field, where a positive number is
an equivalence class containing a sequence
*{x _{1}, x_{2}, x_{3}, ...}*
for which

The real numbers, thus defined, have another important property.
The field is __complete__, which means that every Cauchy sequence
of real numbers has a real limit.

**Theorem 6.1** *The real numbers, as constructed from Cauchy
sequences, is complete.*

*Proof.*
Let *{r _{1}, r_{2}, r_{3}, ...}* be a Cauchy
sequence of real numbers. Then each term

The completeness property may be expressed in other ways. An __upper
bound__ for a set of numbers is just a number greater than, or equal to,
every element of the set.

**Theorem 6.2** *Every nonempty set of real numbers which has an
upper bound has a least upper bound; i.e., an upper bound that is less
than any other upper bound.*

*Proof.* The method of bisection is the simplest proof.
We construct two Cauchy sequences *{x _{1}, x_{2}, x_{3}, ...}*
and

Let *x _{1}* be
an element of the set, and let

At the *n*-th step, *x _{n}* is not an upper bound,
and

These two Cauchy sequences have a common limit, which is the required least upper bound. █

- German translation by Daniel Gruber of akkuschraubercheck.de.