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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[edit]

The multiplicative identity 1 and its additive inverse −1 are always units. Notifyroutechange2. 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 numbersR is R − {0}.

Integers[edit]

In the ring of integersZ, 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 integer1 + √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[edit]

For a commutative ring R, the units of the polynomial ringR[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

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

such that ${\displaystyle a_0}$ is a unit in R.^[4]

Matrix rings[edit]

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[edit]

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:

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${\displaystyle (1-yx{)}^{-1}=\sum _{n\ge 0}(yx{)}^n=1+y\left(\sum _{n\ge 0}(xy{)}^n\right)x=1+y(1-xy{)}^{-1}x.}$

See Hua's identity for similar results.

Group of units[edit]

The units of a ring R form a groupU(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 functorU from the category of rings to the category of groups:

every ring homomorphismf : R → S induces a group homomorphismU(f) : U(R) → U(S), since f maps units to units. Waterfox download.

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

The group scheme${\displaystyle {\mathrm{GL}}_1}$ is isomorphic to the multiplicative group scheme${\displaystyle {\mathbb{G}}_m}$ over any base, so for any commutative ring R, the groups ${\displaystyle {\mathrm{GL}}_1(R)}$ 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: ${\displaystyle {\mathbb{G}}_m(R)\simeq \mathrm{Hom}(\mathbb{Z}[t,t^{-1}],R)}$ 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 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[edit]

Notes[edit]

R/unity3d

1. ^x and −x are not necessarily distinct. For example, in the ring of integers modulo 6, one has 3 = −3 even though 1 ≠ −1.

Citations[edit]

1. ^Dummit & Foote 2004.
2. ^Lang 2002.
3. ^Watkins (2007, Theorem 11.1)
4. ^Watkins (2007, Theorem 12.1)
5. ^Jacobson 2009, § 2.2. Exercise 4.
6. ^Exercise 10 in § 2.2. of Cohn, Paul M. (2003). Further algebra and applications (Revised ed. of Algebra, 2nd ed.). London: Springer-Verlag. ISBN1-85233-667-6. Zbl1006.00001.

Sources[edit]

• Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN0-471-43334-9.
• Jacobson, Nathan (2009). Basic Algebra 1 (2nd ed.). Dover. ISBN978-0-486-47189-1.
• Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Springer. ISBN0-387-95385-X.
• Watkins, John J. (2007), Topics in commutative ring theory, Princeton University Press, ISBN978-0-691-12748-4, MR2330411

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