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

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