Unit (ring theory)

In the branch of abstract algebra known as ring theory, a unit of a ring ${\displaystyle R}$ is any element ${\displaystyle u\in R}$ that has a multiplicative inverse in ${\displaystyle R}$: an element ${\displaystyle v\in R}$ such that

${\displaystyle vu=uv=1}$,

where 1 is the multiplicative identity.^[1][2] The set of units U(R) of a ring forms a group under multiplication.

Less commonly, the term unit is also used to refer to the element 1 of the ring, in expressions like ring with a unit or unit ring, and also e.g. 'unit' matrix. For this reason, some authors call 1 "unity" or "identity", and say that R is a "ring with unity" or a "ring with identity" rather than a "ring with a unit".

Examples[]

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 r^n − 1 is a multiplicative inverse of r. In a nonzero ring, the element 0 is not a unit, so U(R) is not closed under addition. A ring R in which every nonzero element is a unit (that is, U(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 R is R − {0}.

Integers[]

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

The ring of integers in a number field may have more units in general. For example, in the ring Z[1 + √5/ 2] that arises by adjoining the quadratic integer 1 + √5/ 2 to Z, one has

(√5 + 2)(√5 − 2) = 1

in the ring, so √5 + 2 is a unit. (In fact, the unit group of this ring is infinite.^{[citation needed]})

In fact, Dirichlet's unit theorem describes the structure of U(R) precisely: it is isomorphic to a group of the form

${\displaystyle \mathbf {Z} ^{n}\oplus \mu _{R}}$

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 numbers of real embeddings and the number of pairs of complex embeddings of F, respectively.

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

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

Polynomials and power series[]

For a commutative ring R, the units of the polynomial ring R[x] are precisely those polynomials

${\displaystyle p(x)=a_{0}+a_{1}x+\dots a_{n}x^{n}}$

such that ${\displaystyle a_{0}}$ is a unit in R, and the remaining coefficients ${\displaystyle a_{1},\dots ,a_{n}}$ are nilpotent elements, i.e., satisfy ${\displaystyle a_{i}^{N}=0}$ for some N.^[3] In particular, if R is a domain (has no zero divisors), then the units of R[x] agree with the ones of R. The units of the power series ring ${\displaystyle R[[x]]}$ are precisely those power series

${\displaystyle p(x)=\sum _{i=0}^{\infty }a_{i}x^{i}}$

such that ${\displaystyle a_{0}}$ is a unit in R.^[4]

Matrix rings[]

The unit group of the ring M_n(R) of n × n matrices over a ring R is the group GL_n(R) of invertible matrices. For a commutative ring R, an element A of M_n(R) is invertible if and only if the determinant of A is invertible in R. In that case, A⁻¹ is explicitly given by Cramer's rule.

In general[]

For elements x and y in a ring R, if ${\displaystyle 1-xy}$ is invertible, then ${\displaystyle 1-yx}$ is invertible with the inverse ${\displaystyle 1+y(1-xy)^{-1}x}$.^[5] The formula for the inverse 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[]

The units of a ring R form a group U(R) under multiplication, the group of units of R.

Other common notations for U(R) are R^∗, R^×, and E(R) (from the German term Einheit).

A commutative ring is a local ring if R − U(R) is a maximal ideal.

As it turns out, if R − U(R) is an ideal, then it is necessarily a maximal ideal and R is local since a maximal ideal is disjoint from U(R).

If R is a finite field, then U(R) is a cyclic group of order ${\displaystyle |R|-1}$.

The formulation of the group of units defines a functor U from the category of rings to the category of groups:

every ring homomorphism f : R → S induces a group homomorphism U(f) : U(R) → U(S), since f maps units to units.

This functor has a left adjoint which is the integral group ring construction.^[6]

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 ${\displaystyle U(R)}$. Note that the functor ${\displaystyle \mathbb {G} _{m}}$ (that is, ${\displaystyle R\mapsto U(R)}$) 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[]

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^[a] x and −x are associate. 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 U(R) on R via multiplication: Two elements of R are associate if they are in the same U(R)-orbit.

In an integral domain, the set of associates of a given nonzero element has the same cardinality as U(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.

See also[]

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

Notes[]

Citations[]

Sources[]