Unit ring theory

From Exampleproblems

In mathematics, a unit in a (unital) ring R is an invertible element of u, 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.

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 rⁿ = 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