Inalgebra, aunit orinvertible element^[a] of aring is aninvertible element for the multiplication of the ring. That is, an elementu of a ringR is a unit if there existsv inR such that$${\displaystyle vu=uv=1,}$$where1 is themultiplicative identity; the elementv is unique for this property and is called themultiplicative inverse ofu.^[1][2] The set of units ofR forms agroupR^× under multiplication, called thegroup of units orunit group ofR.^[b] Other notations for the unit group areR^∗,U(R), andE(R) (from the German termEinheit).

Less commonly, the termunit is sometimes used to refer to the element1 of the ring, in expressions likering with a unit orunit ring, and alsounit matrix. Because of this ambiguity,1 is more commonly called the "unity" or the "identity" of the ring, and the phrases "ring with unity" or a "ring with identity" may be used to emphasize that one is considering a ring instead of arng.

The multiplicative identity1 and its additive inverse−1 are always units. More generally, anyroot of unity in a ringR is a unit: ifrⁿ = 1, thenrⁿ⁻¹ is a multiplicative inverse ofr.In anonzero ring, theelement 0 is not a unit, soR^× is not closed under addition.A nonzero ringR in which every nonzero element is a unit (that is,R^× =R ∖ {0}) is called adivision ring (or a skew-field). A commutative division ring is called afield. For example, the unit group of the field ofreal numbersR isR ∖ {0}.

In the ring ofintegersZ, the only units are1 and−1.

In the ringZ/nZ ofintegers modulon, the units are the congruence classes(modn) represented by integerscoprime ton. They constitute themultiplicative group of integers modulon.

In the ringZ[√3] obtained by adjoining thequadratic integer√3 toZ, one has(2 +√3)(2 −√3) = 1, so2 +√3 is a unit, and so are its powers, soZ[√3] has infinitely many units.

More generally, for thering of integersR in anumber fieldF,Dirichlet's unit theorem states thatR^× is isomorphic to the group$${\displaystyle {\mathbf{Z}}^n\times {\mu }_R}$$where${\displaystyle {\mu }_R}$ is the (finite, cyclic) group of roots of unity inR andn, therank of the unit group, is$${\displaystyle n=r_1+r_2-1,}$$where${\displaystyle r_1,r_2}$ are the number of real embeddings and the number of pairs of complex embeddings ofF, respectively.

This recovers theZ[√3] example: The unit group of (the ring of integers of) areal quadratic field is infinite of rank 1, since${\displaystyle r_1=2,r_2=0}$.

For a commutative ringR, the units of thepolynomial ringR[x] are the polynomials$${\displaystyle p(x)=a_0+a_{1}x+\cdots +a_n{x}^n}$$such thata₀ is a unit inR and the remaining coefficients${\displaystyle a_1,\dots ,a_n}$ arenilpotent, i.e., satisfy${\displaystyle a_i^N=0}$ for someN.^[4]In particular, ifR is adomain (or more generallyreduced), then the units ofR[x] are the units ofR.The units of thepower series ring${\displaystyle R[[x]]}$ are the power series$${\displaystyle p(x)=\sum _{i=0}^{\mathrm{\infty }}{a}_i{x}^i}$$such thata₀ is a unit inR.^[5]

The unit group of the ringM_n(R) ofn × n matrices over a ringR is the groupGL_n(R) ofinvertible matrices. For a commutative ringR, an elementA ofM_n(R) is invertible if and only if thedeterminant ofA is invertible inR. In that case,A⁻¹ can be given explicitly in terms of theadjugate matrix.

For elementsx andy in a ringR, if${\displaystyle 1-xy}$ is invertible, then${\displaystyle 1-yx}$ is invertible with inverse${\displaystyle 1+y(1-xy{)}^{-1}x}$;^[6] this formula can be guessed, but not proved, by the following calculation in a ring of noncommutative power series:$${\displaystyle (1-yx{)}^{-1}=\sum _{n\ge 0}(yx{)}^n=1+y(\sum _{n\ge 0}(xy{)}^n)x=1+y(1-xy{)}^{-1}x.}$$SeeHua's identity for similar results.

Acommutative ring is alocal ring ifR ∖R^× is amaximal ideal.

As it turns out, ifR ∖R^× is an ideal, then it is necessarily amaximal ideal andR islocal since amaximal ideal is disjoint fromR^×.

IfR is afinite field, thenR^× is acyclic group of order|R| − 1.

Everyring homomorphismf :R →S induces agroup homomorphismR^× →S^×, sincef maps units to units. In fact, the formation of the unit group defines afunctor from thecategory of rings to thecategory of groups. This functor has aleft adjoint which is the integralgroup ring construction.^[7]

Thegroup scheme${\displaystyle {\mathrm{GL}}_1}$ is isomorphic to themultiplicative group scheme${\displaystyle {\mathbb{G}}_m}$ over any base, so for any commutative ringR, the groups${\displaystyle {\mathrm{GL}}_1(R)}$ and${\displaystyle {\mathbb{G}}_m(R)}$ are canonically isomorphic toU(R). Note that the functor${\displaystyle {\mathbb{G}}_m}$ (that is,R ↦U(R)) isrepresentable in the sense:${\displaystyle {\mathbb{G}}_m(R)\simeq \mathrm{Hom}(\mathbb{Z}[t,t^{-1}],R)}$ for commutative ringsR (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${\displaystyle \mathbb{Z}[t,t^{-1}]\to R}$ and the set of unit elements ofR (in contrast,${\displaystyle \mathbb{Z}[t]}$ represents the additive group${\displaystyle {\mathbb{G}}_a}$, theforgetful functor from the category of commutative rings to thecategory of abelian groups).

Suppose thatR is commutative. Elementsr ands ofR are calledassociate if there exists a unitu inR such thatr =us; then writer ~s. In any ring, pairs ofadditive inverse elements^[c]x and−x areassociate, since any ring includes the unit−1. For example, 6 and −6 are associate inZ. In general,~ is anequivalence relation onR.

Associatedness can also be described in terms of theaction ofR^× onR via multiplication: Two elements ofR are associate if they are in the sameR^×-orbit.

In anintegral domain, the set of associates of a given nonzero element has the samecardinality asR^×.

The equivalence relation~ can be viewed as any one ofGreen's semigroup relations specialized to the multiplicativesemigroup of a commutative ringR.

• S-units
• Localization of a ring and a module

• 1 (number)
• Algebraic number theory
• Group theory
• Ring theory
• Algebraic properties of elements

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