Unit (ring theory)

In mathematics, a unit in a 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. The units of R form a group U(R) under multiplication.

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.