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

$X$ must be a commutative group under $\oplus$ . That means:
- $X$ must be closed under $\oplus$ .
- $\oplus$ must be associative.
- $\oplus$ must be commutative.
- $\oplus$ must have an identity, which is usually named $0$ .
- Every $x \in X$ must have an inverse $(- x) \in X$ such that $x \oplus (- x) = 0$ .
$X$ must be a monoid under $\otimes$ . That means:
- $X$ must be closed under $\otimes$ .
- $\otimes$ must be associative.
- $\otimes$ must have an identity, which is usually named $1$ .
$\otimes$ must distribute over $\oplus$ . That means:
- $a \otimes (x \oplus y) = (a \otimes x) \oplus (a \otimes y)$ for all $a, x, y \in X$ .
- $(x \oplus y) \otimes a = (x \otimes a) \oplus (y \otimes a)$ for all $a, x, y \in X$ .

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 $Z$ form a ring under addition and multiplication.

Add more example rings.

[work in progress.]

Notation

Given a ring $R = (X, \oplus, \otimes)$ , we say " $R$ forms a ring under $\oplus$ and $\otimes$ ." $X$ is called the underlying set of $R$ . $\oplus$ is called the "additive operation," $0$ is called the "additive identity", $- x$ is called the "additive inverse" of $x$ . $\otimes$ is called the "multiplicative operation," $1$ 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.]

Ring