In ring theory, a **unit** of a ring is any element that has a multiplicative inverse in : an element such that

- ,

where is the multiplicative identity.^{[1]}^{[2]} The set of units 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 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 additive inverse −1_{R} 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 *r*^{n} = 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 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

where is the (finite, cyclic) group of roots of unity in *R* and *n*, the rank of the unit group is

where 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 .

In the ring **Z**/*n***Z** 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

such that is a unit in *R*, and the remaining coefficients are nilpotent elements, i.e., satisfy 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 are precisely those power series

such that is a unit in *R*.^{[4]}

### 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 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 be a ring. For any in , if is invertible, then is invertible with the inverse .^{[5]} The formula for the inverse can be found as follows: thinking formally, suppose is invertible and that the inverse is given by a geometric series: . Then, manipulating it formally,

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* − 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 .

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

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

## Notes

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

### Citations

**^**Dummit & Foote 2004.**^**Lang 2002.**^**Watkins (2007, Theorem 11.1)**^**Watkins (2007, Theorem 12.1)**^**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