ring

Definition

A ring is a set that has two operations and . They must satisfy the ring axioms:

1. is an abelian group under addition
2. Multiplication is associative

3. Multiplication is closed: for all .
4. Multiplication is distributive:

• left distributive
• right distributive

Morphisms

Morphisms in the category are called ring homomorphisms. A ring homomorphism is a map which satisfies

Relation to subrings and ideals

Let be a ring homomorphism. Then

is a subring of . Similarly,

is an ideal.

This is the main source of “finding” subring and ideals in practice.

Types of rings

Some important classes of rings are

• domain (ring theory)
• division ring
• field

Classes of rings