# Unit (ring theory)

where 1 is the multiplicative identity.^{[1]}^{[2]} The set of units U(*R*) of a ring forms a group under multiplication.

Less commonly, the term *unit* is also used to refer to the element 1 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 "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 ring of integers in a number field may have more units in general. For example, in the ring **Z**[1 + √5/ 2] that arises by adjoining the quadratic integer
1 + √5/ 2 to **Z**, one has

in the ring, so √5 + 2 is a unit. (In fact, the unit group of this ring is infinite.^{[citation needed]})

In fact, Dirichlet's unit theorem describes the structure of U(*R*) precisely: it is isomorphic to a group of the form

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.

For a commutative ring R, the units of the polynomial ring *R*[*x*] are precisely those polynomials

The unit group of the ring M_{n}(*R*) of *n* × *n* matrices over a ring R is the group GL_{n}(*R*) of invertible matrices. For a commutative ring R, an element A of M_{n}(*R*) is invertible if and only if the determinant of A is invertible in R. In that case, *A*^{−1} is explicitly given by Cramer's rule.

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*).

As it turns out, if *R* − U(*R*) is an ideal, then it is necessarily a maximal ideal and *R* is local since a maximal ideal is disjoint from U(*R*).

The formulation of the group of units defines a functor U 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.^{[6]}

Suppose that R is commutative. Elements r and s of R are called *associate* if there exists a unit u in R such that *r* = *us*; then write *r* ∼ *s*. In any ring, pairs of additive inverse elements^{[a]} *x* and −*x* are associate. For example, 6 and −6 are associate in **Z**. In general, ~ is an equivalence relation on R.

Associatedness can also be described in terms of the action of U(*R*) on R via multiplication: Two elements of R are associate if they are in the same U(*R*)-orbit.

In an integral domain, the set of associates of a given nonzero element has the same cardinality as U(*R*).

The equivalence relation ~ can be viewed as any one of Green's semigroup relations specialized to the multiplicative semigroup of a commutative ring R.