Unit (ring theory)

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 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}.

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 $${\displaystyle {\mathbf{Z}}^n\times {\mu }_R}$$ 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 ${\displaystyle r_1=2,r_2=0}$.

Polynomials and power series

For a commutative ring R, the units of the polynomial ring R[x] are the polynomials $${\displaystyle p(x)=a_0+a_{1}x+\cdots +a_n{x}^n}$$ such that a₀ is a unit in R and the remaining coefficients ${\displaystyle a_1,\dots ,a_n}$ are nilpotent, i.e., satisfy ${\displaystyle a_i^N=0}$ for some N. In particular, if R is a domain (or more generally reduced), then the units of R[x] are the units of R. The units of the power series ring ${\displaystyle R[[x]]}$ are the power series $${\displaystyle p(x)=\sum _{i=0}^{\mathrm{\infty }}{a}_i{x}^i}$$ such that a₀ is a unit in R.

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 ${\displaystyle 1+y(1-xy{)}^{-1}x}$; this formula can be guessed, but not proved, by the following calculation in a ring of noncommutative power series: $${\displaystyle (1-yx{)}^{-1}=\sum _{n\ge 0}(yx{)}^n=1+y(\sum _{n\ge 0}(xy{)}^n)x=1+y(1-xy{)}^{-1}x.}$$ See Hua's identity for similar results.