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.

Title	unit
Canonical name	Unit
Date of creation	2013-03-22 11:56:28
Last modified on	2013-03-22 11:56:28
Owner	drini (3)
Last modified by	drini (3)
Numerical id	15
Author	drini (3)
Entry type	Definition
Classification	msc 16B99
Synonym	unital
Related topic	Associates
Related topic	Prime
Related topic	Ring
Related topic	UnitsOfQuadraticFields
Defines	algebraic unit