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
Less commonly, the term unit is sometimes used to refer to the element 1 of the ring, in expressions like ring with a unit or unit ring, and also unit matrix. Because of this ambiguity, 1 is more commonly called the "unity" or the "identity" of the ring, and the phrases "ring with unity" or a "ring with identity" may be used to emphasize that one is considering a ring instead of a rng.
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 of integers Z, the only units are 1 and −1.
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
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 .
For a commutative ring R, the units of the polynomial ring R[x] are the polynomials
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.
For elements x and y in a ring R, if is invertible, then is invertible with inverse ;[6] this formula can be guessed, but not proved, by the following calculation in a ring of noncommutative power series:
A commutative ring is a local ring if R − R× is a maximal ideal.
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×.
If R is a finite field, then R× is a cyclic group of order .
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]
The group scheme is isomorphic to the multiplicative group scheme over any base, so for any commutative ring R, the groups and are canonically isomorphic to . Note that the functor (that is, ) is representable in the sense: 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, represents the additive group , the forgetful functor from the category of commutative rings to the category of abelian groups).
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[c] 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.