Unit (ring theory)

In mathematics, a unit in a (unital) ring R is an element u such that there is v in R with

uv = vu = 1_R.

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.

Group of units

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

The orbits of U(R) acting on R by multiplication 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. For example in the ring Z of integers n and −n are associates.

Examples

Any root of unity is a unit. 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 fact that is the source for the unit terminology — which shouldn't be confused with the 'unit' of unital rings.

One can check that U is a functor from the category of rings, to the category of groups: a ring homomorphism must map units to units. It has a left adjoint, the integral group ring construction.