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

$${\displaystyle vu=uv=1,}$$

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.

Examples

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 r^n − 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}.

Integer ring

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.

Ring of integers of a number field

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

$${\displaystyle \mathbf {Z} ^{n}\times \mu _{R))$$

${\displaystyle \mu _{R))$

$${\displaystyle n=r_{1}+r_{2}-1,}$$

${\displaystyle r_{1},r_{2))$

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 ${\displaystyle r_{1}=2,r_{2}=0}$.

Polynomials and power series

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

$${\displaystyle p(x)=a_{0}+a_{1}x+\dots +a_{n}x^{n))$$

${\displaystyle a_{0))$

${\displaystyle a_{1},\dots ,a_{n))$

${\displaystyle a_{i}^{N}=0}$

${\displaystyle R[[x]]}$

$${\displaystyle p(x)=\sum _{i=0}^{\infty }a_{i}x^{i))$$

${\displaystyle a_{0))$

Matrix rings

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.

In general

For elements x and y in a ring R, if ${\displaystyle 1-xy}$ is invertible, then ${\displaystyle 1-yx}$ is invertible with inverse ${\displaystyle 1+y(1-xy)^{-1}x}$;^[6] this formula can be guessed, but not proved, by the following calculation in a ring of noncommutative power series:

$${\displaystyle (1-yx)^{-1}=\sum _{n\geq 0}(yx)^{n}=1+y\left(\sum _{n\geq 0}(xy)^{n}\right)x=1+y(1-xy)^{-1}x.}$$

Group of units

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 ${\displaystyle |R|-1}$.

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.^[7]

The group scheme ${\displaystyle \operatorname {GL} _{1))$ is isomorphic to the multiplicative group scheme ${\displaystyle \mathbb {G} _{m))$ over any base, so for any commutative ring R, the groups ${\displaystyle \operatorname {GL} _{1}(R)}$ and ${\displaystyle \mathbb {G} _{m}(R)}$ are canonically isomorphic to ${\displaystyle U(R)}$. Note that the functor ${\displaystyle \mathbb {G} _{m))$ (that is, ${\displaystyle R\mapsto U(R)}$) is representable in the sense: ${\displaystyle \mathbb {G} _{m}(R)\simeq \operatorname {Hom} (\mathbb {Z} [t,t^{-1}],R)}$ 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 ${\displaystyle \mathbb {Z} [t,t^{-1}]\to R}$ and the set of unit elements of R (in contrast, ${\displaystyle \mathbb {Z} [t]}$ represents the additive group ${\displaystyle \mathbb {G} _{a))$, the forgetful functor from the category of commutative rings to the category of abelian groups).

Associatedness

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.