Unit (ring theory)

In mathematics, an invertible element or a unit in a (unital) ring $R$ is any element $u$ that has an inverse element in the multiplicative monoid of $R$ , i.e. an element $v$ such that

uv = vu = 1 R

, where

1 R

is the multiplicative identity.^[1]^[2]

The set of units of any ring is closed under multiplication (the product of two units is again a unit), and forms a group for this operation. It never contains the element 0 (except in the case of the zero ring), and is therefore not closed under addition; its complement however might be a group under addition, which happens if and only if the ring is a local ring.

The term unit is also used to refer to the identity element $1 R$ of the ring, in expressions like ring with a unit or unit ring, and also e.g. 'unit' matrix. For this reason, some authors call $1 R$ "unity" or "identity", and say that $R$ is a "ring with unity" or a "ring with identity" rather than a "ring with a unit".

The multiplicative identity $1 R$ and its opposite $-1 R$ are always units. Hence, pairs of additive inverse elements^[3] $x$ and $- x$ are always associated.

Examples

In any ring, 1 is a unit. More generally, any root of unity in a ring $R$ is a unit: if $r n = 1$ , then $r n - 1$ is a multiplicative inverse of $r$ . On the other hand, 0 is never a unit. A ring R is a field (possibly non-commutative, also known as a skew field or division ring) if and only if $U(R) = R ∖ {0}$ , where U(R) is the group of units of R. For example, the units of the real numbers $R$ are $R ∖ {0$ }. Thus, for any ring R, there is an inclusion

{\text{roots of unity }}\subset U(R)\subset R\backslash \{0\}.

Integers

In the ring of integers $Z$ , the only units are $+1$ and $-1$ .

Rings of integers $R={\mathfrak {O}}_{F}$ in a number field F have, in general, more units. For example,

(\sqrt 5 + 2)(\sqrt 5 - 2) = 1

in the ring $Z [1 + \sqrt 5 / 2]$ , and in fact the unit group of this ring is infinite.

In fact, Dirichlet's unit theorem describes the structure of $U(R)$ precisely: it is isomorphic to a group of the form

\mathbf {Z} ^{n}\oplus \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 numbers of real embeddings and the number of pairs of complex embeddings of F, respectively.

This recovers the above example: the unit group of (the ring of integers of) a real quadratic field is infinite of rank 1, since $r_{1}=2,r_{2}=0$ .

In the ring $Z / n Z$ 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 $n$ .

Polynomials and power series

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

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

such that $a_{0}$ is a unit in R, and the remaining coefficients $a_{1},\dots ,a_{n}$ are nilpotent elements, i.e., satisfy $a_{i}^{N}=0$ for some N.^[4] In particular, if R is a domain (has no zero divisors), then the units of R[x] agree with the ones of R. The units of the power series ring $R[[x]]$ are precisely those power series

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

such that $a_{0}$ is a unit in R.^[5]

Matrix rings

The unit group of the ring $M n (F)$ of $n \times n$ matrices over a field $F$ is the group $GL n (F)$ of invertible matrices.

Group of units

The units of a ring $R$ form a group $U(R)$ under multiplication, the group of units of $R$ . Other common notations for $U(R)$ are $R *$ , $R \times$ , and $E(R)$ (from the German term Einheit).

This defines a functor $U$ from the category of rings to the category of groups: every ring homomorphism $f : R \to S$ induces a group homomorphism $U(f) : U(R) \to U(S)$ , since $f$ maps units to units. This functor has a left adjoint which is the integral group ring construction.

Associatedness

In a commutative unital ring $R$ , the group of units $U(R)$ acts on $R$ via multiplication. The orbits of this action are called sets of associates; in other words, there is an equivalence relation ∼ on $R$ called associatedness such that

r \sim s

means that there is a unit $u$ with $r = us$ .

In an integral domain the cardinality of an equivalence class of associates is the same as that of $U(R)$ .

References

↑ Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9.
↑ Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Springer. ISBN 0-387-95385-X.
↑ In a ring, the additive inverse of a non-zero element can equal to the element itself.
↑ Watkins (2007, Theorem 11.1)
↑ Watkins (2007, Theorem 12.1)

Watkins, John J. (2007), Topics in commutative ring theory, Princeton University Press, ISBN 978-0-691-12748-4, MR 2330411

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