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In the branch of abstract algebra known as ring theory, a **unit** of a ring is any element that has a multiplicative inverse in : an element such that

- ,

where 1 is the multiplicative identity.^{[1]}^{[2]} The set of units of R forms a group *R*^{×} under multiplication, called the **group of units** or **unit group** of R.^{[a]} Other notations for the unit group are *R*^{∗}, U(*R*), and E(*R*) (from the German term *Einheit*).

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".

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 *r ^{n}* = 1, then

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

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.

In the ring **Z**[√3] obtained by adjoining the quadratic integer √3 to **Z**, one has (2 + √3)(2 - √3) = 1, so 2 + √3 is a unit, and so are its powers, so **Z**[√3] has infinitely many units.

More generally, for the ring of integers R in a number field F, Dirichlet's unit theorem states that *R*^{×} is isomorphic to the group

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

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

This recovers the **Z**[√3] example: The unit group of (the ring of integers of) a real quadratic field is infinite of rank 1, since .

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

such that is a unit in R and the remaining coefficients are nilpotent, i.e., satisfy for some *N*.^{[4]}
In particular, if R is a domain, then the units of *R*[*x*] are the units of R.
The units of the power series ring are the power series

such that is a unit in R.^{[5]}

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} can be given explicitly in terms of the adjugate matrix.

For elements x and y in a ring R, if is invertible, then is invertible with inverse ;^{[6]} this formula can be guessed, but not proved, by the following calculation in a ring of noncommutative power series:

See Hua's identity for similar results.

A commutative ring is a local ring if *R* − *R*^{×} is a maximal ideal.

As it turns out, if *R* − *R*^{×} is an ideal, then it is necessarily a maximal ideal and *R* is local since a maximal ideal is disjoint from *R*^{×}.

If R is a finite field, then *R*^{×} is a cyclic group of order .

Every ring homomorphism *f* : *R* → *S* induces a group homomorphism *R*^{×} → *S*^{×}, since f maps units to units. In fact, the formation of the unit group defines a functor from the category of rings to the category of groups. This functor has a left adjoint which is the integral group ring construction.^{[7]}

The group scheme is isomorphic to the multiplicative group scheme over any base, so for any commutative ring R, the groups and are canonically isomorphic to . Note that the functor (that is, ) is representable in the sense: 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 and the set of unit elements of *R* (in contrast, represents the additive group , the forgetful functor from the category of commutative rings to the category of abelian groups).

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^{[b]} *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 *R*^{×} on R via multiplication: Two elements of R are associate if they are in the same *R*^{×}-orbit.

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

**^**The notation*R*^{×}, introduced by André Weil, is commonly used in number theory, where unit groups arise frequently.^{[3]}The symbol × is a reminder that the group operation is multiplication. Also, a superscript × is not frequently used in other contexts, whereas a superscript * often denotes dual.**^**x and −*x*are not necessarily distinct. For example, in the ring of integers modulo 6, one has 3 = −3 even though 1 ≠ −1.

**^**Dummit & Foote 2004.**^**Lang 2002.**^**Weil 1974.**^**Watkins (2007, Theorem 11.1)**^**Watkins (2007, Theorem 12.1)**^**Jacobson 2009, § 2.2. Exercise 4.**^**Exercise 10 in § 2.2. of Cohn, Paul M. (2003).*Further algebra and applications*(Revised ed. of Algebra, 2nd ed.). London: Springer-Verlag. ISBN 1-85233-667-6. Zbl 1006.00001.

- 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 - Weil, André (1974).
*Basic number theory*. Grundlehren der mathematischen Wissenschaften. Vol. 144 (3rd ed.). Springer-Verlag. ISBN 978-3-540-58655-5.