nLab group of units

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 (units of the ring), and whose group operation is the multiplication in $R$.

Remark

$GL_1(R)$ is an affine variety (in fact an affine algebraic group) over $R$, namely $\{(x, y) \in R^2: x y = 1\}$.

This leads us to the following alternative perspective:

Definition

In a category with finite limits, with $R$ a ring object therein, the group of units of $R$ is the equalizer of the two maps $m, c_1: R \times R \to R$, where $m$ is the ring multiplication and $c_1$ is the constant map with value the multiplicative identity.

Cf. Example below.

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 multiplicative group of the ring of integers modulo $n$ is the multiplicative group of integers modulo n.

Example

The group of units of the ring of adeles $\mathbb{A}$ is the group of ideles. The topology on the idele group $\mathbb{I}$ arises by considering $\mathbb{I}$ as an affine variety in $\mathbb{A}^2$ as above, and giving it the subspace topology. This is not the subspace topology induced by the inclusion $\mathbb{I} \hookrightarrow\mathbb{A}$ into the ring of adeles.

Example

The group of units of the $p$-adic integers $\mathbb{Z}_p$ fits in an exact sequence

$$1 \to 1 + p \mathbb{Z}_p \hookrightarrow \mathbb{Z}_p^\times \to (\mathbb{Z}/(p))^\times \to 1$$

where the quotient is isomorphic to the cyclic group $\mathbb{Z}/(p-1)$ (see root of unity) and the kernel is, at least when $p \gt 2$, isomorphic to the additive group $\mathbb{Z}_p$. Explicitly, for such $p$ the formal exponential map $\exp(x) = \sum_{n \geq 0} \frac{x^n}{n!}$ converges when $x \in p \mathbb{Z}_p$ and maps $p \mathbb{Z}_p$ isomorphically onto the multiplicative group $1 + p \mathbb{Z}_p$. The formal logarithm $\log(x) = \sum_{n \geq 1} \frac{(-1)^{n-1} (x - 1)^n}{n}$ is also convergent for $x \in 1 + p \mathbb{Z}_p$ and provides the inverse.

By Hensel's lemma, the group of units $\mathbb{Z}_p^\times$ has $(p-1)^{th}$ roots of unity and therefore the exact sequence above splits. This splitting descends to the quotient ring $\mathbb{Z}/(p^n)$ and its group of units, giving an isomorphism $GL_1(\mathbb{Z}/(p^n)) \cong \mathbb{Z}/(p^{n-1}) \oplus \mathbb{Z}/(p-1)$.

Related concepts

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

• multiplicative submonoid of cancellative elements

• ∞-group of units

• multiplicative group

• exponential sequence, Kummer sequence

References

For instance:

• Anthony Knapp, p. 143 in: Basic Algebra, Springer (2006) [doi:10.1007/978-0-8176-4529-8, pdf]