nLab
group of units

Definition

For $R$ a ring, its group of units, denoted $R^{\times }$ or ${\mathrm{GL}}_1(R)$, 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

Remark

There is an adjunction

$$(R[-]⊣(-{)}^{\times }):{\mathrm{Alg}}_R\stackrel{\stackrel{R[-]}{\leftarrow }}{\underset{(-{)}^{\times }}{\to }}\mathrm{Grp}$$(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.

Related concepts

• group of units/multiplicative group, Picard group, Brauer group

• ∞-group of units