## You are here

Homeunit

## Primary tabs

# unit

Let $R$ be a ring with multiplicative identity $1$. We say that $u\in R$ is an unit (or unital) if $u$ divides $1$ (denoted $u\mid 1$). That is, there exists an $r\in R$ such that $1=ur=ru$.

Notice that $r$ will be the multiplicative inverse (in the ring) of $u$, so we can characterize the units as those elements of the ring having multiplicative inverses.

In the special case that $R$ is the ring of integers of an algebraic number field $K$, the units of $R$ are sometimes called the algebraic units of $K$ (and also the units of $K$). They are determined by Dirichlet’s unit theorem.

Defines:

algebraic unit

Keywords:

Ring, Factorization

Related:

Associates, Prime, Ring, UnitsOfQuadraticFields

Synonym:

unital

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

16B99*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections