# Unit (ring theory)

In mathematics, a **unit** in a (unital) ring *R* is an invertible element of *R*, i.e. an element *u* such that there is a *v* in *R* with

*uv*=*vu*= 1_{R}, where 1_{R}is the multiplicative identity element.

That is, *u* is an *invertible* element of the multiplicative monoid of *R*.

Unfortunately, the term *unit* is also used to refer to the identity element 1_{R} 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_{R} "unity", and say that *R* is a "ring with unity" rather than "ring with a unit".)

## Group of units

The units of *R* form a group *U*(*R*) under multiplication, the **group of units** of *R*. The group of units *U*(*R*) is sometimes also denoted *R*^{*} or *R*^{×}.

In a commutative unital ring *R*, the group of units *U*(*R*) acts on *R* via multiplication. The orbits of this action are called sets of *associates*; in other words, there is an equivalence relation ~ on *R* called *associatedness* such that

*r*~*s*

means that there is a unit *u* with *r* = *us*.

One can check that *U* is a functor 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.

A ring *R* is a field if and only if *R*^{*} = *R* \ {0}.

## Examples

- In the ring of integers,
**Z**, the units are ±1. The associates are pairs*n*and −*n*.

- Any root of unity is a unit in any unital ring
*R*. (If*r*is a root of unity, and*r*^{n}= 1, then*r*^{−1}=*r*^{n − 1}is also an element of*R*by closure under multiplication.) In algebraic number theory, Dirichlet's unit theorem shows the existence of many units in most rings of algebraic integers. For example, we have (√5 + 2)(√5 − 2) = 1.

- In the ring
*M*(*n*,**F**) of*n*×*n*matrices over some field**F**the units are exactly the invertible matrices.