Ring, Field and Integral Domain

Rings

Ring

A ring is a set equipped with two laws of composition and , called addition and multiplication, satisfying the following axioms:

• is an abelian group with identity , called zero;
• The multiplication is associative: for all ;
• Distributivity: and for all .

If has an identity with respect to multiplication, we say that is unital.
A ring is commutative if the multiplication is commutative.

e.g.

• is the trivial ring with , it is sometimes called the zero ring;
• The integers is a commutative unital ring, any is a commutative unital ring;
• The set of 2×2 matrices with real entries is a non-commutative unital ring.
  

Subring

A subset in a ring is called a subring if is closed under addition, subtraction, multiplication and contains (if is unital).

e.g.

• The trivial ring is not a subring of any other nontrivial rings, even though itself is a ring. The point is

Unit

An element of a ring is called a unit if it is invertible with respect to multiplication. The set of invertible elements is a group called the the group of units in and denotes .

Proposition

For any (non-unital) ring , is a unital ring with multiplicative identity under the multiplication

where for any , is a shorthand for ( times) if , if , and if . (any abelian group is a $\Z$-module)

Proof This is easy to check.

From the above proposition, we can see that any ring can be enlarged into a unital ring. So we can just focus on unital rings most of the time. From now on,

Attention

We shall sometimes just call commutative unital rings “rings”, and emphasize the other cases when necessary.

Field and Domain

Zero Divisor & Integral Domain

A zero divisor in a ring is a non-zero element such that for some non-zero . A ring without zero divisors is called an integral domain. In other words, an integral domain is a ring in which the product of any two non-zero elements is non-zero.

Cancellation Law

An integral domain satisfies the cancellation law: if and then .

Proof Suppose and . Then . Since and is an integral domain, .

Field

A field is a ring in which and every non-zero element is invertible, that is .

e.g. The rational numbers is a field.

Proposition

Any field is an integral domain.

Proof Suppose , can be either or a unit, if it is not , then .

Proposition

is a field if and only if is a prime.

Proof Suppose is prime, then every non-zero element satisfies by Fermat’s Little Theorem. Therefore . Conversely, if is not prime, then there exists such that but , so is not even an integral domain.

Proposition

Every finite integral domain is a field.

Proof Suppose is an integral domain with finitely many elements. Let be a non-zero element. Consider the function defined by . Since the cancellation law holds in an integral domain, is injective. Because is finite, must also be surjective (ref. proposition). Therefore, there exists some such that , showing that every non-zero element has a multiplicative inverse. Hence, is a field.

Given any integral domain, we can construct the smallest field containing it, called the fraction field:

The Field of Fractions

Suppose is an integral domain. Consider equip with the following relation:
This is an equivalence relation. Let denote the quotient. We denote the equivalence classes by .

Universal Property of the Field of Fractions

is the smallest field containing . More precisely, for any field and any injective ring homomorphism , there exists a unique field homomorphism such that the following diagram commute:

Proof

In general, if a ring is not an integral domain, we can still attempt to construct a “field of fractions”, this is called the [localisation](https://www.wikiwand.com/en/Localization_(commutative_algebra).

Ordered Field

Ordered Field

An ordered field is a field along with a subset of , called the positive subset, with the following properties:

• if , then or or .
• if , then .
• if , then and .

Equivalently, a field together with a total order on is an ordered field if the order satisfies the following properties for all :

• .
• .
  

Proposition

The positive subset is closed under multiplicative inverse. i.e. Suppose is an ordered field with positive subset . Then and for all .