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
v u = u v = 1 , {\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.
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
Z n × μ R {\displaystyle \mathbf {Z} ^{n}\times \mu _{R}}
where μ R {\displaystyle \mu _{R}} is the (finite, cyclic) group of roots of unity in R and n, the rank of the unit group, is
n = r 1 + r 2 − 1 , {\displaystyle n=r_{1}+r_{2}-1,}
where r 1 , r 2 {\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 r 1 = 2 , r 2 = 0 {\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
p ( x ) = a 0 + a 1 x + ⋯ + a n x n {\displaystyle p(x)=a_{0}+a_{1}x+\dots +a_{n}x^{n}}
such that a0 is a unit in R and the remaining coefficients a 1 , … , a n {\displaystyle a_{1},\dots ,a_{n}} are nilpotent, i.e., satisfy a i N = 0 {\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 R [ [ x ] ] {\displaystyle R[[x]]} are the power series
p ( x ) = ∑ i = 0 ∞ a i x i {\displaystyle p(x)=\sum _{i=0}^{\infty }a_{i}x^{i}}
such that a0 is a unit in R.
Matrix rings 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.
In general For elements x and y in a ring R, if 1 − x y {\displaystyle 1-xy} is invertible, then 1 − y x {\displaystyle 1-yx} is invertible with inverse 1 + y ( 1 − x y ) − 1 x {\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:
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