Unit (ring theory)


In mathematics, an invertible element or a unit in a ring with identity is any element that has an inverse element in the multiplicative monoid of, i.e. an element such that
The set of units of any ring is closed under multiplication, and forms a group for this operation. It never contains the element 0, 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 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".
The multiplicative identity and its additive inverse are always units. Hence, pairs of additive inverse elements and 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, then is a multiplicative inverse of r.
On the other hand, 0 is never a unit. A ring R is called a skew-field if, where U is the group of units of R. A commutative skew-field is called a field. For example, the units of the real numbers are.

Integers

In the ring of integers, the only units are and.
Rings of integers in a number field F have, in general, more units. For example,
in the ring, and 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 is the 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 a real quadratic field is infinite of rank 1, since.
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 R, the units of the polynomial ring R are precisely those polynomials
such that is a unit in R, and the remaining coefficients are nilpotent elements, i.e., satisfy for some N. In particular, if R is a domain, then the units of R 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.

Matrix rings

The unit group of the ring of matrices over a commutative ring is the group general linear group| 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. 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 form a group under multiplication, the group of units of. Other common notations for are,, and .
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.
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.

Associatedness

In a commutative unital ring, the group of units acts on via multiplication. The orbits of this action are called sets of ; in other words, there is an equivalence relation ∼ on called associatedness such that
means that there is a unit with.
In an integral domain the cardinality of an equivalence class of associates is the same as that of.