# Ring

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.

• $$\oplus$$ 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 $$x \oplus (-x) = 0$$.

1. $$X$$ 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 $$1$$.

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 $$\mathbb{Z}$$ 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.]