Unit (ring theory)

In the branch of abstract algebra known as ring theory, a unit of a ring is any element that has a multiplicative inverse in : an element such that

,

where 1 is the multiplicative identity.[1][2] The set of units U(R) of a ring forms a group under multiplication.

Less commonly, the term unit is also used to refer to the element 1 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 "unity" or "identity", and say that R is a "ring with unity" or a "ring with identity" rather than a "ring with a unit".

ExamplesEdit

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 U(R) is not closed under addition. A ring R in which every nonzero element is a unit (that is, U(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}.

IntegersEdit

In the ring of integers Z, the only units are 1 and −1.

The ring of integers in a number field may have more units in general. For example, in the ring Z[1 + 5/ 2] that arises by adjoining the quadratic integer 1 + 5/ 2 to Z, one has

(5 + 2)(5 − 2) = 1

in the ring, so 5 + 2 is a unit. (In fact, the unit group of this ring is infinite.[citation needed])

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

 

where   is the (finite, cyclic) group of roots of unity in R and n, the rank of the unit group is

 

where   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  .

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.

Polynomials and power seriesEdit

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

 

such that   is a unit in R, and the remaining coefficients   are nilpotent elements, i.e., satisfy   for some N.[3] 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   are precisely those power series

 

such that   is a unit in R.[4]

Matrix ringsEdit

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 is explicitly given by Cramer's rule.

In generalEdit

For elements x and y in a ring R, if   is invertible, then   is invertible with the inverse  .[5] The formula for the inverse can be guessed, but not proved, by the following calculation in a ring of noncommutative power series:

 

See Hua's identity for similar results.

Group of unitsEdit

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×, and E(R) (from the German term Einheit).

A commutative ring is a local ring if R − U(R) is a maximal ideal.

As it turns out, if R − U(R) is an ideal, then it is necessarily a maximal ideal and R is local since a maximal ideal is disjoint from U(R).

If R is a finite field, then U(R) is a cyclic group of order  .

The formulation of the group of units defines a functor U from the category of rings to the category of groups:

every ring homomorphism f : RS induces a group homomorphism U(f) : U(R) → U(S), since f maps units to units.

This functor has a left adjoint which is the integral group ring construction.[6]

The group scheme   is isomorphic to the multiplicative group scheme   over any base, so for any commutative ring R, the groups   and   are canonically isomorphic to  . Note that the functor   (that is,  ) is representable in the sense:   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   and the set of unit elements of R (in contrast,   represents the additive group  , the forgetful functor from the category of commutative rings to the category of abelian groups).

AssociatednessEdit

Suppose that R is commutative. Elements r and s of R are called associate if there exists a unit u in R such that r = us; then write rs. In any ring, pairs of additive inverse elements[a] x and x are associate. For example, 6 and −6 are associate in Z. In general, ~ is an equivalence relation on R.

Associatedness can also be described in terms of the action of U(R) on R via multiplication: Two elements of R are associate if they are in the same U(R)-orbit.

In an integral domain, the set of associates of a given nonzero element has the same cardinality as U(R).

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

See alsoEdit

NotesEdit

  1. ^ x and x are not necessarily distinct. For example, in the ring of integers modulo 6, one has 3 = −3 even though 1 ≠ −1.

CitationsEdit

  1. ^ Dummit & Foote 2004.
  2. ^ Lang 2002.
  3. ^ Watkins (2007, Theorem 11.1)
  4. ^ Watkins (2007, Theorem 12.1)
  5. ^ Jacobson 2009, § 2.2. Exercise 4.
  6. ^ Exercise 10 in § 2.2. of Cohn, Paul M. (2003). Further algebra and applications (Revised ed. of Algebra, 2nd ed.). London: Springer-Verlag. ISBN 1-85233-667-6. Zbl 1006.00001.

SourcesEdit