Unit (ring theory)

Not to be confused with Unit ring.

In mathematics, an invertible element or a unit in a (unital) ring R is any element u that has an inverse element in the multiplicative monoid of R, i.e. an element v such that

uv = vu = 1_R, where 1_R is the multiplicative identity.

The set of units of any ring is closed under multiplication (the product of two units is again a unit), and forms a group for this operation. It never contains the element 0 (except in the case of the zero ring), and is therefore not closed under addition; its complement however might be a group under addition, which happens if and only if the ring is a local ring.

The term unit is also used to refer to the identity element 1_R 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_R "unity" or "identity", and say that R is a "ring with unity" or a "ring with identity" rather than a "ring with a unit".

The multiplicative identity 1_R and its opposite −1_R are always units. Hence, pairs of additive inverse elements x and −x are always associated.

Examples

In any ring, 1 is a unit. 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. On the other hand, 0 is never a unit (except in the zero ring). A ring R is called a skew-field (or a division ring) if U(R) = R - {0}, where U(R) is the group of units of R (see below). A commutative skew-field is called a field. For example, the units of the real numbers R are R - {0}.

Integers

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

Rings of integers $R={\mathfrak {O}}_{F}$ in a number field F have, in general, more units. For example,

(√5 + 2)(√5 − 2) = 1

in the ring Z[1 + √5/2], and in fact the unit group of this ring is infinite.

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

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

where $\mu _{R}$ is the (finite, cyclic) group of roots of unity in R and n, the rank of the unit group is

$n=r_{1}+r_{2}-1,$

where $r_{1},r_{2}$ 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 $r_{1}=2,r_{2}=0$.

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

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

such that $a_{0}$ is a unit in R, and the remaining coefficients $a_{1},\dots ,a_{n}$ are nilpotent elements, i.e., satisfy $a_{i}^{N}=0$ for some N. 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 $R[[x]]$ are precisely those power series

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

such that $a_{0}$ is a unit in R.

Matrix rings

The unit group of the ring M_n(R) of n × n matrices over a commutative ring R (for example, a field) is the group GL_n(R) of invertible matrices.

An element of the matrix ring $\operatorname {M} _{n}(R)$ is invertible if and only if the determinant of the element is invertible in R, with the inverse explicitly given by Cramer's rule.

In general

Let $R$ be a ring. For any $x,y$ in $R$, if $1-xy$ is invertible, then $1-yx$ is invertible with the inverse $1+y(1-xy)^{-1}x$. The formula for the inverse can be found as follows: thinking formally, suppose $1-yx$ is invertible and that the inverse is given by a geometric series: $(1-yx)^{-1}=\sum _{0}^{\infty }(yx)^{n}$. Then, manipulating it formally,

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

See also Hua's identity for a similar type of 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 $R-\operatorname {U} (R)$ is a maximal ideal (as it turns out, if $R-\operatorname {U} (R)$ is an ideal, then it is necessarily a maximal ideal and R is local since a maximal ideal is disjoint from $\operatorname {U} (R)$.)

If R is a finite field, then $\operatorname {U} (R)$ is a cyclic group of order the cardinality of R minus one.

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.

Associatedness

Main article: Green's relations § The H and D relations

In a commutative unital ring R, the group of units U(R) acts on R via multiplication. The orbits of this action are called sets of associates; in other words, there is an equivalence relation ∼ on R called associatedness such that

r ∼ s

means that there is a unit u with r = us.

In an integral domain the cardinality of an equivalence class of associates is the same as that of U(R).

See also

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

References

• Jacobson, Nathan (2009), Basic Algebra 1 (2nd ed.), Dover, ISBN 978-0-486-47189-1
• Watkins, John J. (2007), Topics in commutative ring theory, Princeton University Press, ISBN 978-0-691-12748-4, MR 2330411