Group of units[edit]
A commutative ring is a local ring if R ∖ R× is a maximal ideal.
As it turns out, if R ∖ R× is an ideal, then it is necessarily a maximal ideal and R is local since a maximal ideal is disjoint from R×.
If R is a finite field, then R× is a cyclic group of order |R| − 1.
Every ring homomorphism f : R → S induces a group homomorphism R× → S×, since f maps units to units. In fact, the formation of the unit group defines a functor from the category of rings to the category of groups. This functor has a left adjoint which is the integral group ring construction.[7]
The group scheme is isomorphic to the multiplicative group scheme over any base, so for any commutative ring R, the groups and are canonically isomorphic to U(R). Note that the functor (that is, R ↦ U(R)) is representable in the sense: for commutative rings R (this for instance follows from the aforementioned adjoint relation with the group ring construction). Explicitly this means that there is a natural bijection between the set of the ring homomorphisms and the set of unit elements of R (in contrast, represents the additive group , the forgetful functor from the category of commutative rings to the category of abelian groups).
![{\displaystyle \mathbb {G} _{m}(R)\simeq \operatorname {Hom} (\mathbb {Z} [t,t^{-1}],R)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c7385d9d048f7923fa85690937000b3e2b015cd)
![{\displaystyle \mathbb {Z} [t,t^{-1}]\to R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/097602f428d486a9ea02003085c0fddf8b8a4ee6)
![{\displaystyle \mathbb {Z} [t]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55e65744a826d17b7e9fa264dc341006b9be5d55)
Associatedness[edit]
Suppose that R is commutative. Elements r and s of R are called associate if there exists a unit u in R such that r = us; then write r ~ s. In any ring, pairs of additive inverse elements[c] x and −x are associate, since any ring includes the unit −1. For example, 6 and −6 are associate in Z. In general, ~ is an equivalence relation on R.
Associatedness can also be described in terms of the action of R× on R via multiplication: Two elements of R are associate if they are in the same R×-orbit.
In an integral domain, the set of associates of a given nonzero element has the same cardinality as R×.
The equivalence relation ~ can be viewed as any one of Green's semigroup relations specialized to the multiplicative semigroup of a commutative ring R.