# Unit (ring theory)

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 = 1R, where 1R 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 1R 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 1R "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. (U(R) is sometimes also denoted 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. For example, in the ring Z of integers, n and −n are associates.

## Examples

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 = rn − 1 is also an 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 fact, that is the source for the unit terminology — which shouldn't be confused with the 'unit' (unity) of unital rings.

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. It has a left adjoint, the integral group ring construction.de:Einheit (Mathematik) fr:Élément inversible

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