nLab group of units

Context

Algebra

higher algebra

universal algebra

group theory

Contents

Definition

For $R$ a ring, its group of units, denoted ${R}^{×}$ or ${\mathrm{GL}}_{1}\left(R\right)$, 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

$\left(R\left[-\right]⊣\left(-{\right)}^{×}\right):{\mathrm{Alg}}_{R}\stackrel{\stackrel{R\left[-\right]}{←}}{\underset{\left(-{\right)}^{×}}{\to }}\mathrm{Grp}$(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 $R\left[-\right]$ forms the group algebra over $R$ and where $\left(-{\right)}^{×}$ assigns to an $R$-algebra its group of units.