# nLab group of units

### Context

#### Algebra

higher algebra

universal algebra

group theory

# Contents

## Definition

###### Definition

For $R$ a ring, its group of units, denoted $R^\times$ or $GL_1(R)$, is the group whose elements are the elements of $R$ that are invertible under the product, and whose group operation is the multiplication in $R$.

## Properties

### Relation to the multiplicative group

###### Proposition

The group of units of $R$ is equivalently the collection of morphisms from $Spec R$ into the group of units $\mathbb{G}_m$

$GL_1(R) = R^\times \simeq Hom(Spec R, \mathbb{G}_m) \,.$

### Relation to the group ring

###### Remark

There is an adjunction

$(R[-]\dashv (-)^\times) \colon Alg_R \stackrel{\overset{R[-]}{\leftarrow}}{\underset{(-)^\times}{\to}} Grp$

between the category of associative algebras over $R$ and that of groups, where $R[-]$ forms the group algebra over $R$ and where $(-)^\times$ assigns to an $R$-algebra its group of units.

## Examples

###### Example

The group of units of the ring of adeles is the group of ideles.

Revised on November 21, 2013 10:43:14 by Urs Schreiber (188.200.54.65)