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group of units

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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.

Remark

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

This leads us to the following alternative perspective:

Definition

In a category with finite limits, with RR a ring object therein, the group of units of RR is the equalizer of the two maps m,c 1:R×RRm, c_1: R \times R \to R, where mm is the ring multiplication and c 1c_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 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 multiplicative group of the ring of integers modulo nn is the multiplicative group of integers modulo n.

Example

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

Example

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

11+p p p ×(/(p)) ×11 \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 /(p1)\mathbb{Z}/(p-1) (see root of unity) and the kernel is, at least when p>2p \gt 2, isomorphic to the additive group p\mathbb{Z}_p. Explicitly, for such pp the formal exponential map exp(x)= n0x nn!\exp(x) = \sum_{n \geq 0} \frac{x^n}{n!} converges when xp px \in p \mathbb{Z}_p and maps p pp \mathbb{Z}_p isomorphically onto the multiplicative group 1+p p1 + p \mathbb{Z}_p. The formal logarithm log(x)= n1(1) n1(x1) nn\log(x) = \sum_{n \geq 1} \frac{(-1)^{n-1} (x - 1)^n}{n} is also convergent for x1+p px \in 1 + p \mathbb{Z}_p and provides the inverse.

By Hensel's lemma, the group of units p ×\mathbb{Z}_p^\times has (p1) th(p-1)^{th} roots of unity and therefore the exact sequence above splits. This splitting descends to the quotient ring /(p n)\mathbb{Z}/(p^n) and its group of units, giving an isomorphism GL 1(/(p n))/(p n1)/(p1)GL_1(\mathbb{Z}/(p^n)) \cong \mathbb{Z}/(p^{n-1}) \oplus \mathbb{Z}/(p-1).

Last revised on September 27, 2018 at 05:01:09. See the history of this page for a list of all contributions to it.