# 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.]