Rings, Integral Domains and Fields

Philip J. Erdelsky

March 24, 2007

Please e-mail comments, corrections and additions to the webmaster at pje@efgh.com.

1. Rings

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 {0, 1, ..., M-1}, where addition and multiplication are modulo M, is a commutative ring with a unit. We shall see some noncommutative rings later.

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.

2. Integral Domains

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 {0, 1, ..., M-1}, where addition and multiplication are modulo M, is an integral domain if M is prime.

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.

3. Fields

An integral domain is a field if every nonzero element x has a reciprocal x^-1 such that xx^-1 = x^-1x = 1. Notice that the reciprocal is just the inverse under multiplication; therefore, the nonzero elements of a field are a commutative group under multiplication. The real numbers are one familiar field, and the ring Z_p is a field if p is prime. In fact, it is fairly easy to prove that any finite integral domain is a field.

Division in a field is defined in the usual way:

x / y = x y^-1,

where the denominator y must be nonzero.

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
(-a)/b = a/(-b) = -(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.

4. Fields of Quotients and the Rational Numbers

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) if ad = 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.

5. Ordered Integral Domains

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.

6. The Field of Real Numbers

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 m² = 2n², which is impossible because the prime factor 2 would appear an even number of times in the left member and an odd number of times in the right member. Hence every rational number is either less than the square root of 2 or greater than the square root of 2, but never equal to the square root of 2.

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 {x₁, x₂, x₃, ...} of rational numbers is a Cauchy sequence if for every positive e there is an integer n such that ∣x_i - x_j∣ < e whenever i>n and j>n.

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 {x₁, x₂, x₃, ...} and {y₁, y₂, y₃, ...} are equivalent if their term-by-term difference {x₁ - y₁, x₂ - y₂, x₃ - y₃, ...} approaches zero. It is easily shown that this is indeed an equivalence relation.

The equivalence classes are the real numbers.

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

{x₁, x₂, x₃, ...} + {y₁, y₂, y₃, ...} = {x₁ + y₁, x₂ + y₂, x₃ + y₃, ...}.
{x₁, x₂, x₃, ...} {y₁, y₂, y₃, ...} = {x₁ y₁, x₂ y₂, x₃ y₃, ...}.

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 {x₁, x₂, x₃, ...} for which x_i > e for all i > n for some positive e and some integer n. It is easily seen to be archimedean.

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 field of real numbers, as constructed from Cauchy sequences, is complete.

Proof. Let {r₁, r₂, r₃, ...} be a Cauchy sequence of real numbers. Then each term r_i is represented by an equivalence class of Cauchy sequences of rational numbers. Pick one, and find the first term in it such that difference in absolute values of subsequent terms in it will always be less than 1/i. The equivalence class of the sequence of such first terms is the limit of the original sequence of real numbers. █

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 {x₁, x₂, x₃, ...} and {y₁, y₂, y₃, ...} as follows.

Let x₁ be an element of the set, and let y₁ be an upper bound.

At the n-th step, x_n is not an upper bound, and y_n is an upper bound. Let m = (x_n + y_n)/2. Then if m is not an upper bound, let x_n+1 = m and y_n+1 = y_n. If m is an upper bound, let x_n+1 = x_n and y_n+1 = m.

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