Unit (ring theory)

In algebra, a unit or invertible element 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 $${\displaystyle vu=uv=1,}$$ where 1 is the multiplicative identity; the element v is unique for this property and is called the multiplicative inverse of u. The set of units of R forms a group R^× under multiplication, called the group of units or unit group of R. Other notations for the unit group are R^∗, U(R), and E(R) (from the German term Einheit).

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 $${\displaystyle \mathbf {Z} ^{n}\times \mu _{R}}$$ where ${\displaystyle \mu _{R}}$ is the (finite, cyclic) group of roots of unity in R and n, the rank of the unit group, is $${\displaystyle n=r_{1}+r_{2}-1,}$$ where ${\displaystyle r_{1},r_{2}}$ are the number of real embeddings and the number of pairs of complex embeddings of F, respectively.

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