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 $$vu = uv = 1,$$ 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 $R×$, $×$, and $$ (from the German term Einheit).

Less commonly, the term unit is sometimes used to refer to the element $R^{∗}$ of the ring, in expressions like ring with a unit or unit ring, and also unit matrix. Because of this ambiguity, $U(R)$ 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 $E(R)$ and its additive inverse $1$ are always units. More generally, any root of unity in a ring $R$ is a unit: if $1$, then $1$ is a multiplicative inverse of $r$. In a nonzero ring, the element 0 is not a unit, so $−1$ is not closed under addition. A nonzero ring $R$ in which every nonzero element is a unit (that is, $r^{n} = 1$) 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 $r^{n−1}$ is $R×$.

Integer ring
In the ring of integers $R× = R &setminus; \{0\}$, the only units are $R$ and $R &setminus; \{0\}$.

In the ring $Z$ of integers modulo $n$, the units are the congruence classes $1$ 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 $−1$ obtained by adjoining the quadratic integer $Z/nZ$ to $(mod n)$, one has $Z[√3]$, so $√3$ is a unit, and so are its powers, so $Z$ has infinitely many units.

More generally, for the ring of integers $R$ in a number field $F$, Dirichlet's unit theorem states that $(2 + √3)(2 − √3) = 1$ is isomorphic to the group $$\mathbf Z^n \times \mu_R$$ where $$\mu_R$$ is the (finite, cyclic) group of roots of unity in $R$ and $n$, the rank of the unit group, is $$n = r_1 + r_2 -1, $$ where $$r_1, r_2$$ are the number of real embeddings and the number of pairs of complex embeddings of $F$, respectively.

This recovers the $2 + √3$ example: The unit group of (the ring of integers of) a real quadratic field is infinite of rank 1, since $$r_1=2, r_2=0$$.

Polynomials and power series
For a commutative ring $R$, the units of the polynomial ring $Z[√3]$ are the polynomials $$p(x) = a_0 + a_1 x + \dots + a_n x^n$$ such that $R×$ is a unit in $R$ and the remaining coefficients $$a_1, \dots, a_n$$ are nilpotent, i.e., satisfy $$a_i^N = 0$$ for some $Z[√3]$. 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 $$Rx$$ are the power series $$p(x)=\sum_{i=0}^\infty a_i x^i$$ such that $a_{0}$ is a unit in $R$.

Matrix rings
The unit group of the ring $N$ of $R[x]$ matrices over a ring $R$ is the group $a_{0}$ 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, $n × n$ can be given explicitly in terms of the adjugate matrix.

In general
For elements $x$ and $y$ in a ring $R$, if $$1 - xy$$ is invertible, then $$1 - yx$$ is invertible with inverse $$1 + y(1-xy)^{-1}x$$; this formula can be guessed, but not proved, by the following calculation in a ring of noncommutative power series: $$(1-yx)^{-1} = \sum_{n \ge 0} (yx)^n = 1 + y \left(\sum_{n \ge 0} (xy)^n \right)x = 1 + y(1-xy)^{-1}x.$$ See Hua's identity for similar results.

Group of units
A commutative ring is a local ring if $GL_{n}(R)$ is a maximal ideal.

As it turns out, if $M_{n}(R)$ is an ideal, then it is necessarily a maximal ideal and $A−1$ is local since a maximal ideal is disjoint from $R &setminus; R×$.

If $R$ is a finite field, then $R &setminus; R×$ is a cyclic group of order $R$.

Every ring homomorphism $R×$ induces a group homomorphism $R×$, 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.

The group scheme $$\operatorname{GL}_1$$ is isomorphic to the multiplicative group scheme $$\mathbb{G}_m$$ over any base, so for any commutative ring $R$, the groups $$\operatorname{GL}_1(R)$$ and $$\mathbb{G}_m(R)$$ are canonically isomorphic to $|R| − 1$. Note that the functor $$\mathbb{G}_m$$ (that is, $f : R → S$) is representable in the sense: $$\mathbb{G}_m(R) \simeq \operatorname{Hom}(\mathbb{Z}[t, t^{-1}], R)$$ 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 $$\mathbb{Z}[t, t^{-1}] \to R$$ and the set of unit elements of $R$ (in contrast, $$\mathbb{Z}[t]$$ represents the additive group $$\mathbb{G}_a$$, 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  if there exists a unit $u$ in $R$ such that $R× → S×$; then write $U(R)$. In any ring, pairs of additive inverse elements $R ↦ U(R)$ and $r = us$ are associate, since any ring includes the unit $r ~ s$. For example, 6 and −6 are associate in $−x$. In general, $3 = −3$ is an equivalence relation on $x$.

Associatedness can also be described in terms of the action of $1 ≠ −1$ on $R$ via multiplication: Two elements of $R$ are associate if they are in the same $x$-orbit.

In an integral domain, the set of associates of a given nonzero element has the same cardinality as $−x$.

The equivalence relation $−1$ can be viewed as any one of Green's semigroup relations specialized to the multiplicative semigroup of a commutative ring $R$.