[summary: A ring is a kind of Algebraic structure which we obtain by considering groups as being "things with addition" and then endowing them with a multiplication operation which must interact appropriately with the pre-existing addition. Terminology varies across sources; we will take "ring" to refer to "commutative ring with $~$1$~$".]

[summary(Technical): A ring $~$R$~$ 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$~$. Terminology varies across sources; our rings will have both operations commutative and will have an Identity element under multiplication, denoted $~$1$~$.]

A ring $~$R$~$ 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 $~$xy$~$ when $~$\otimes$~$ can be inferred from context. The ten ring axioms (which govern the behavior of $~$\oplus$~$ and $~$\otimes$~$) are as follows:

1. $~$X$~$ must be a commutative group under $~$\oplus$~$. That means:

• $~$X$~$ must be closed under $~$\oplus$~$.
• $~$\oplus$~$ must be [associative_function associative].
• $~$\oplus$~$ must be [commutative_function 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$~$.

1. $~$X$~$ must be a monoid under $~$\otimes$~$. That means:

• $~$X$~$ must be closed under $~$\otimes$~$.
• $~$\otimes$~$ must be [associative_function associative].
• $~$\otimes$~$ must have an identity, which is usually named $~$1$~$.

1. $~$\otimes$~$ must [distributive_property 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 [distributive_property 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 $~$\mathbb{Z}$~$ form a ring under addition and multiplication.

[fixme: 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

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

Interpretations, Visualizations, and Applications

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