# Unit (ring theory)

In mathematics, an **invertible element** or a **unit** in a (unital) ring R is any element u that has an inverse element in the multiplicative monoid of R, i.e. an element v such that

*uv*=*vu*= 1_{R}, where 1_{R}is the multiplicative identity.^{[1]}^{[2]}

The set of units of any ring is closed under multiplication (the product of two units is again a unit), and forms a group for this operation. It never contains the element 0 (except in the case of the zero ring), and is therefore not closed under addition; its complement however might be a group under addition, which happens if and only if the ring is a local ring.

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" or "identity", and say that R is a "ring with unity" or a "ring with identity" rather than a "ring with a unit".

The multiplicative identity 1_{R} and its opposite −1_{R} are always units. Hence, pairs of additive inverse elements^{[3]} *x* and −*x* are always associated.

## Group of units

The units of a ring R form a group U(*R*) under multiplication, the **group of units** of R. Other common notations for U(*R*) are *R*^{∗}, *R*^{×}, and E(*R*) (from the German term *Einheit*).

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.

In an integral domain the cardinality of an equivalence class of associates is the same as that of U(*R*).

A ring R is a division ring if and only if U(*R*) = *R* ∖ {0}.

## Examples

- In the ring of integers
**Z**, the only units are +1 and −1. - In the ring
**Z**/*n***Z**of integers modulo n, the units are the congruence classes (mod*n*) represented by integers coprime to n. They constitute the multiplicative group of integers modulo*n*. - Any root of unity in a ring R is a unit. (If
*r*^{n}= 1, then*r*^{n − 1}is a multiplicative inverse of r.) - If
*R*is the ring of integers in a number field, Dirichlet's unit theorem implies that the unit group of*R*is a finitely generated abelian group. For example, we have (√5 + 2)(√5 − 2) = 1 in the ring**Z**[1 + √5/2], and in fact the unit group of this ring is infinite. In general, the unit group of (the ring of integers of) a real quadratic field is infinite (of rank 1). - The unit group of the ring M
_{n}(*F*) of*n*×*n*matrices over a field F is the group GL_{n}(*F*) of invertible matrices. - The units of the real numbers
**R**are**R**∖ {0}.

## References

- ↑ Dummit, David S.; Foote, Richard M. (2004).
*Abstract Algebra*(3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9. - ↑ Lang, Serge (2002).
*Algebra*. Graduate Texts in Mathematics. Springer. ISBN 0-387-95385-X. - ↑ In a ring, the additive inverse of a non-zero element can equal to the element itself.