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Unit (ring theory)

Inalgebra, a unitorinvertible element^[a] of a ring is an invertible element for the multiplication of the ring. That is, an element u of a ring R is a unit if there exists vinR such that $${\displaystyle vu=uv=1,}$$ where 1 is the multiplicative identity; the element v is unique for this property and is called the multiplicative inverseofu.^[1][2] The set of units of R forms a group R^× under multiplication, called the group of unitsorunit groupofR.^[b] Other notations for the unit group are R^∗, U(R), and E(R) (from the German term Einheit).

Less commonly, the term unit is sometimes used to refer to the element 1 of the ring, in expressions like ring with a unitorunit ring, and also unit 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 a rng.

Examples[edit]

The multiplicative identity 1 and its additive inverse −1 are always units. More generally, any root of unity in a ring R is a unit: if rⁿ = 1, then rⁿ⁻¹ is a multiplicative inverse of r. In a nonzero ring, the element 0 is not a unit, so R^× is not closed under addition. A nonzero ring R in which every nonzero element is a unit (that is, R^× = R ∖ {0}) is called a division ring (or a skew-field). A commutative division ring is called a field. For example, the unit group of the field of real numbers RisR ∖ {0}.

Integer ring[edit]

In the ring of integers Z, the only units are 1 and −1.

In the ring Z/nZofintegers modulo n, the units are the congruence classes (mod n) represented by integers coprimeton. They constitute the multiplicative group of integers modulo n.

Ring of integers of a number field[edit]

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

More generally, for the ring of integers R in a number field F, Dirichlet's unit theorem states that R^× 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 in R and n, the rank 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 of F, respectively.

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

Polynomials and power series[edit]

For a commutative ring R, the units of the polynomial ring R[x] are the polynomials $${\displaystyle p($$x)=a_{0}+a_{1}x+\dots +a_{n}x^{n}} such that a₀ is a unit in R and the remaining coefficients ${\displaystyle a_{1},\dots ,a_{n}}$ are nilpotent, i.e., satisfy ${\displaystyle a_{i}^{N}=0}$ for some N.^[4] In particular, if R is a domain (or more generally reduced), then the units of R[x] are the units of R. The units of the power series ring ${\displaystyle R[[$x]]} are the power series $${\displaystyle p($$x)=\sum _{i=0}^{\infty }a_{i}x^{i}} such that a₀ is a unit in R.^[5]

Matrix rings[edit]

The unit group of the ring M_n(R)ofn × n matrices over a ring R is the group GL_n(R)ofinvertible matrices. For a commutative ring R, an element AofM_n(R) is invertible if and only if the determinantofA is invertible in R. In that case, A⁻¹ can be given explicitly in terms of the adjugate matrix.

In general[edit]

For elements x and y in a ring R, 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: $${\displaystyle (1-yx)^{-1}=\sum _{n\geq 0}($$yx)^{n}=1+y\left(\sum _{n\geq 0}(xy)^{n}\right)x=1+y(1-xy)^{-1}x.} See Hua's identity for similar results.

Group of units[edit]

Acommutative ring is a local ringifR ∖ R^× is a maximal ideal.

As it turns out, if R ∖ R^× is an ideal, then it is necessarily a maximal ideal and Rislocal since a maximal ideal is disjoint from R^×.

IfR 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 ${\displaystyle \operatorname {GL} _{1}}$ is isomorphic to the multiplicative group scheme ${\displaystyle \mathbb {G} _{m}}$ over any base, so for any commutative ring R, the groups ${\displaystyle \operatorname {GL} _{1}($R)} and ${\displaystyle \mathbb {G} _{m}($R)} are canonically isomorphic to U(R). Note that the functor ${\displaystyle \mathbb {G} _{m}}$ (that is, R ↦ U(R)) is representable in the sense: ${\displaystyle \mathbb {G} _{m}($R)\simeq \operatorname {Hom} (\mathbb {Z} [t,t^{-1}],R)} 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 ${\displaystyle \mathbb {Z} [t,t^{-1}]\to R}$ and the set of unit elements of R (in contrast, ${\displaystyle \mathbb {Z} [$t]} represents the additive group ${\displaystyle \mathbb {G} _{a}}$, the forgetful functor from the category of commutative rings to the category of abelian groups).

Associatedness[edit]

Suppose that R is commutative. Elements r and sofR are called associate if there exists a unit uinR 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 relationonR.

Associatedness can also be described in terms of the actionofR^×onR 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 cardinalityasR^×.

The equivalence relation ~ can be viewed as any one of Green's semigroup relations specialized to the multiplicative semigroup of a commutative ring R.

See also[edit]

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

Notes[edit]

Citations[edit]

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