Unit (ring theory)

Summary

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 of R forms a group R× under multiplication, called the group of units or unit group of R.[a] Other notations for the unit group are R, U(R), and E(R) (from the German term Einheit).

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".

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

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

where 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 .

Polynomials and power series

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

such that is a unit in R and the remaining coefficients are nilpotent, i.e., satisfy for some N.[4] In particular, if R is a domain, then the units of R[x] are the units of R. The units of the power series ring are the power series

such that is a unit in R.[5]

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 is invertible, then is invertible with inverse ;[6] this formula 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 units

A commutative ring is a local ring if RR× is a maximal ideal.

As it turns out, if RR× is an ideal, then it is necessarily a maximal ideal and R is local since a maximal ideal is disjoint from R×.

If R is a finite field, then R× is a cyclic group of order .

Every ring homomorphism f : RS induces a group homomorphism R×S×, 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.[7]

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).

Associatedness

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[b] 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 R× on R via multiplication: Two elements of R are associate if they are in the same R×-orbit.

In an integral domain, the set of associates of a given nonzero element has the same cardinality as 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 also

Notes

  1. ^ The notation R×, introduced by André Weil, is commonly used in number theory, where unit groups arise frequently.[3] The symbol × is a reminder that the group operation is multiplication. Also, a superscript × is not frequently used in other contexts, whereas a superscript * often denotes dual.
  2. ^ x and x are not necessarily distinct. For example, in the ring of integers modulo 6, one has 3 = −3 even though 1 ≠ −1.

Citations

  1. ^ Dummit & Foote 2004.
  2. ^ Lang 2002.
  3. ^ Weil 1974.
  4. ^ Watkins (2007, Theorem 11.1)
  5. ^ Watkins (2007, Theorem 12.1)
  6. ^ Jacobson 2009, § 2.2. Exercise 4.
  7. ^ 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.

Sources