In algebra, a unit or invertible element[a] of a ring is an invertible element for the multiplication of the ring.
That is, an element u of a ring R is a unit if there exists v in R such that
[1][2] The set of units of R forms a group R× under multiplication, called the group of units or unit group of R.[b] Other notations for the unit group are R∗, U(R), and E(R) (from the German term Einheit).
The multiplicative identity 1 and its additive inverse −1 are always units.
More generally, any root of unity in a ring R is a unit: if rn = 1, then rn−1 is a multiplicative inverse of r. In a nonzero ring, the element 0 is not a unit, so R× is not closed under addition.
A nonzero ring R in which every nonzero element is a unit (that is, R× = R ∖ {0}) is called a division ring (or a skew-field).
A commutative division ring is called a field.
For example, the unit group of the field of real numbers R is R ∖ {0}.
In the ring Z/nZ 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
is the (finite, cyclic) group of roots of unity in R and n, the rank of the unit group, is
This recovers the Z[√3] example: The unit group of (the ring of integers of) a real quadratic field is infinite of rank 1, since
such that a0 is a unit in R.[5] The unit group of the ring Mn(R) of n × n matrices over a ring R is the group GLn(R) of invertible matrices.
For a commutative ring R, an element A of Mn(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.
;[6] this formula can be guessed, but not proved, by the following calculation in a ring of noncommutative power series:
See Hua's identity for similar results.
If R is a finite field, then R× is a cyclic group of order |R| − 1.
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.
is isomorphic to the multiplicative group scheme
over any base, so for any commutative ring R, the groups
for commutative rings R (this for instance follows from the aforementioned adjoint relation with the group ring construction).
Explicitly this means that there is a natural bijection between the set of the ring homomorphisms
and the set of unit elements of R (in contrast,
, the forgetful functor from the category of commutative rings to the category of abelian groups).
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[c] x and −x are associate, since any ring includes the unit −1.
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.