Unit (ring theory)

In the branch of abstract algebra known as ring theory, a unit of a ring formula_1 is any element formula_2 that has a multiplicative inverse in formula_1: an element formula_4 such that
where is the multiplicative identity. The set of units of a ring forms a group under multiplication.
Less commonly, the term "unit" is also used to refer to the element of the ring, in expressions like "ring with a unit" or ', and also e.g. unit' matrix". For this reason, some authors call "unity" or "identity", and say that is a "ring with unity" or a "ring with identity" rather than a "ring with a unit".
Examples.
The multiplicative identity and its additive inverse are always units. More generally, any root of unity in a ring "R" is a unit: if , then is a multiplicative inverse of .
In a nonzero ring, the element 0 is not a unit, so is not closed under addition.
A ring in which every nonzero element is a unit (that is, ) 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 is }.
Integers.
In the ring of integers , the only units are and .
The ring of integers in a number field may have more units in general. For example, in the ring that arises by adjoining the quadratic integer to , one has
in the ring, so is a unit. (In fact, the unit group of this ring is infinite.)
In fact, Dirichlet's unit theorem describes the structure of precisely: it is isomorphic to a group of the form
where formula_7 is the (finite, cyclic) group of roots of unity in and , the rank of the unit group is
where formula_9 are the numbers of real embeddings and the number of pairs of complex embeddings of , 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 formula_10.
In the ring of integers modulo, the units are the congruence classes represented by integers coprime to . They constitute the multiplicative group of integers modulo.
Polynomials and power series.
For a commutative ring , the units of the polynomial ring are precisely those polynomials
such that formula_12 is a unit in , and the remaining coefficients formula_13 are nilpotent elements, i.e., satisfy formula_14 for some "N".
In particular, if "R" is a domain (has no zero divisors), then the units of agree with the ones of .
The units of the power series ring formula_15 are precisely those power series
such that formula_12 is a unit in .
Matrix rings.
The unit group of the ring of matrices over a ring is the group of invertible matrices. For a commutative ring , an element of is invertible if and only if the determinant of is invertible in . In that case, is explicitly given by Cramer's rule.
In general.
For elements and in a ring , if formula_18 is invertible, then formula_19 is invertible with the inverse formula_20. 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.
Group of units.
The units of a ring form a group under multiplication, the "group of units" of .
Other common notations for are , , and (from the German term ).
A commutative ring is a local ring if is a maximal ideal.
As it turns out, if is an ideal, then it is necessarily a maximal ideal and "R" is local since a maximal ideal is disjoint from .
If is a finite field, then is a cyclic group of order formula_22.
The formulation of the group of units defines a functor from the category of rings to the category of groups:
every ring homomorphism induces a group homomorphism , since maps units to units.
This functor has a left adjoint which is the integral group ring construction.
The group scheme formula_23 is isomorphic to the multiplicative group scheme formula_24 over any base, so for any commutative ring , the groups formula_25 and formula_26 are canonically isomorphic to formula_27. Note that the functor formula_24 (that is, formula_29) is representable in the sense: formula_30 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 formula_31 and the set of unit elements of "R" (in contrast, formula_32 represents the additive group formula_33, the forgetful functor from the category of commutative rings to the category of abelian groups).
Associatedness.
Suppose that is commutative. Elements and of are called "" if there exists a unit in such that ; then write . In any ring, pairs of additive inverse elements and are associate. For example, 6 and −6 are associate in . In general, is an equivalence relation on .
Associatedness can also be described in terms of the action of on via multiplication: Two elements of are associate if they are in the same -orbit.
In an integral domain, the set of associates of a given nonzero element has the same cardinality as .
The equivalence relation can be viewed as any one of Green's semigroup relations specialized to the multiplicative semigroup of a commutative ring .