nLab
group of units

Context

Algebra

Group Theory

Contents

Definition

Definition

For RR a ring, its group of units, denoted R ×R^\times or GL 1(R)GL_1(R), is the group whose elements are the elements of RR that are invertible under the product, and whose group operation is the multiplication in RR.

Properties

Relation to the multiplicative group

Proposition

The group of units of RR is equivalently the collection of morphisms from SpecRSpec R into the group of units 𝔾 m\mathbb{G}_m

GL 1(R)=R ×Hom(SpecR,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[]() ×):Alg R() ×R[]Grp (R[-]\dashv (-)^\times) \colon Alg_R \stackrel{\overset{R[-]}{\leftarrow}}{\underset{(-)^\times}{\to}} Grp

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

Examples

Example

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

Revised on May 26, 2014 02:43:10 by Urs Schreiber (31.55.33.159)