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: for all .

If has identity with respect to multiplication we say that is unital.

Commutative Ring

A ring is commutative if the multiplication is commutative.

e.g. The integers is a commutative unital ring.

Attention

We will restrict to commutative unital rings and just call them rings.

Subring

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

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 .

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.

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.

Cancellation Law

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

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

Divisibility in Rings

Let be a ring. An element divides if there exists with .

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 .