Written by Nate Soares, et al. last updated

A ring is a triple where is a set and and are binary operations subject to the ring axioms. We write for the application of to , which must be defined, and similarly for . It is standard to abbreviate as when can be inferred from context. The ten ring axioms (which govern the behavior of and ) are as follows:

  1. must be a commutative group under . That means:
    • must be closed under .
    • must be associative.
    • must be commutative.
    • must have an identity, which is usually named .
    • Every must have an inverse such that .
  2. must be a monoid under . That means:
    • must be closed under .
    • must be associative.
    • must have an identity, which is usually named .
  3. must distribute over . That means:
    • for all .
    • for all .

Though the axioms are many, the idea is simple: A ring is a commutative group equipped with an additional operation, under which the ring is a monoid, and the two operations play nice together (the monoid operation distributes over the group operation).

A ring is an algebraic structure. To see how it relates to other algebraic structures, refer to the tree of algebraic structures.

Examples

The integers form a ring under addition and multiplication.

Add more example rings.
[work in progress.]

Notation

Given a ring , we say " forms a ring under and ." is called the underlying set of . is called the "additive operation," is called the "additive identity", is called the "additive inverse" of . is called the "multiplicative operation," is called the "multiplicative identity", and a ring does not necessarily have multiplicative inverses.

Basic properties

Add the basic properties of rings.
[work in progress.]

Interpretations, Visualizations, and Applications

Add (links to) interpretations, visualizations, and applications.
[work in progress.]