group of units



Group Theory




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.


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:


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 1 below.


Relation to the multiplicative group


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


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.



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.


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).

Revised on July 13, 2015 03:23:18 by Urs Schreiber (