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.]

In­ter­pre­ta­tions, Vi­su­al­iza­tions, and Applications

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

Children:

Parents: