In
algebra, a unit of a
ring is an
invertible element for the multiplication of the ring. That is, an element of a ring is a unit if there exists in such that
where is the
multiplicative identity
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
; the element is unique for this property and is called the
multiplicative inverse of . The set of units of forms a
group under multiplication, called the group of units or unit group of . Other notations for the unit group are , , and (from the German term ).
Less commonly, the term ''unit'' is sometimes used to refer to the element of the ring, in expressions like ''ring with a unit'' or ''unit ring'', and also
unit matrix. Because of this ambiguity, is more commonly called the "unity" or the "identity" of the ring, and the phrases "ring with unity" or a "ring with identity" may be used to emphasize that one is considering a ring instead of a
rng.
Examples
The multiplicative identity and its additive inverse are always units. More generally, any
root of unity in a ring 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 nonzero 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 .
Integer ring
In the ring of
integers , the only units are and .
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 .
Ring of integers of a number field
In the ring obtained by adjoining the
quadratic integer to , one has , so is a unit, and so are its powers, so has infinitely many units.
More generally, for the
ring of integers
In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often deno ...
in a
number field ,
Dirichlet's unit theorem states that is isomorphic to the group
where
is the (finite, cyclic) group of roots of unity in and , the
rank of the unit group, is
where
are the number of real embeddings and the number of pairs of complex embeddings of , respectively.
This recovers the example: The unit group of (the ring of integers of) a
real quadratic field is infinite of rank 1, since
.
Polynomials and power series
For a commutative ring , the units of the
polynomial ring are the polynomials
such that
is a unit in and the remaining coefficients
are
nilpotent, i.e., satisfy
for some ''N''.
In particular, if is a
domain, then the units of are the units of .
The units of the
power series ring are the power series
such that
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, can be given explicitly in terms of the
adjugate matrix.
In general
For elements and in a ring , if
is invertible, then
is invertible with inverse
; 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.
Group of units
A
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
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
.
Every
ring homomorphism induces a
group homomorphism , since 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.
[Exercise 10 in § 2.2. of ]
The
group scheme is isomorphic to the
multiplicative group scheme
In mathematics, a group scheme is a type of object from algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups ha ...
over any base, so for any commutative ring , the groups
and
are canonically isomorphic to
. Note that the functor
(that is,
) is representable in the sense:
for commutative rings (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 (in contrast,