# Unit (ring theory)

where 1 is the multiplicative identity.^{[1]}^{[2]} The set of units of R forms a group *R*^{×} under multiplication, called the **group of units** or **unit group** of R.^{[a]} Other notations for the unit group are *R*^{∗}, U(*R*), and E(*R*) (from the German term *Einheit*).

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".

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.

In the ring **Z**[√3] obtained by adjoining the quadratic integer √3 to **Z**, one has (2 + √3)(2 - √3) = 1, so 2 + √3 is a unit, and so are its powers, so **Z**[√3] has infinitely many units.

More generally, for the ring of integers R in a number field F, Dirichlet's unit theorem states that *R*^{×} is isomorphic to the group

For a commutative ring R, the units of the polynomial ring *R*[*x*] are the 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} can be given explicitly in terms of the adjugate matrix.

As it turns out, if *R* − *R*^{×} is an ideal, then it is necessarily a maximal ideal and *R* is local since a maximal ideal is disjoint from *R*^{×}.

Every ring homomorphism *f* : *R* → *S* induces a group homomorphism *R*^{×} → *S*^{×}, since f maps units to units. In fact, the formation of the unit group defines a functor from the category of rings to the category of groups. This functor has a left adjoint which is the integral group ring construction.^{[7]}

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^{[b]} *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 *R*^{×} on R via multiplication: Two elements of R are associate if they are in the same *R*^{×}-orbit.

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