# Ring

https://arbital.com/p/algebraic_ring

by Nate Soares May 9 2016 updated Jul 28 2016

[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.]