# Unit (ring theory)

In 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

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

where ${\displaystyle 1_{R}}$ is the multiplicative identity.[1][2] The set of units ${\displaystyle U(R)}$ of a ring forms a group under multiplication, because it is closed under multiplication. (The product of two units is again a unit.) 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 1R 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 1R "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 1R and its additive inverse −1R are always units. Hence, pairs of additive inverse elements[a] x and x are always associated.

## Examples

1 is a unit in any ring. 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. 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 ${\displaystyle R={\mathfrak {O}}_{F}}$ 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

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

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 Mn(R) of n × n matrices over a commutative ring R (for example, a field) is the group GLn(R) of invertible matrices.

An element of the matrix ring ${\displaystyle \operatorname {M} _{n}(R)}$ 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 ${\displaystyle R}$ be a ring. For any ${\displaystyle x,y}$ in ${\displaystyle R}$, 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 found as follows: thinking formally, suppose ${\displaystyle 1-yx}$ is invertible and that the inverse is given by a geometric series: ${\displaystyle (1-yx)^{-1}=\sum _{0}^{\infty }(yx)^{n}}$. Then, manipulating it formally,

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

## 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 : RS 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

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

rs

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).

## Notes

1. ^ In a ring, the additive inverse of a non-zero element can equal the element itself.

### Citations

1. ^
2. ^
3. ^ Watkins (2007, Theorem 11.1)
4. ^ Watkins (2007, Theorem 12.1)
5. ^ Jacobson 2009, § 2.2. Exercise 4.

## Sources

• Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9.
• Jacobson, Nathan (2009). Basic Algebra 1 (2nd ed.). Dover. ISBN 978-0-486-47189-1.
• Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Springer. ISBN 0-387-95385-X.
• Watkins, John J. (2007), Topics in commutative ring theory, Princeton University Press, ISBN 978-0-691-12748-4, MR 2330411