Unit (ring Theory)

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## Examples

### Integers

### Polynomials and power series

### Matrix rings

### In general

## Group of units

## Associatedness

## See also

## Notes

### Citations

## Sources

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Unit Ring Theory

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 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".

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 *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 numbers **R** is **R** - {0}.

In the ring of integers **Z**, 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**[] that arises by adjoining the quadratic integer to **Z**, one has

- ( + 2)( - 2) = 1

in the ring, so + 2 is a unit. (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.

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]}

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.

For elements x and y in a ring R, if is invertible, then is invertible with the inverse .^{[5]} The formula for the inverse can be guessed, but not proved, by the following calculation in a ring of noncommutative power series:

See Hua's identity for similar results.

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.^{[6]}

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^{[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.

**^**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.**^**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

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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