Unit (ring theory)

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search

In ring theory, a unit of a ring is any element that has a multiplicative inverse in : an element such that


where is the multiplicative identity.[1][2] The set of units of a ring forms a group under multiplication, because it is closed under multiplication. (The product of two units is again a unit.) 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 1R 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 1R "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 1R and its additive inverse −1R are always units. Hence, pairs of additive inverse elements[a] x and x are always associated.


1 is a unit in any ring. 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. On the other hand, 0 is never a unit (except in the zero ring). A ring R is called a skew-field (or a division ring) if U(R) = R - {0}, where U(R) is the group of units of R (see below). A commutative skew-field is called a field. For example, the units of the real numbers R are R - {0}.


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

Rings of integers in a number field F have, in general, more units. For example,

(5 + 2)(5 − 2) = 1

in the ring Z[1 + 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

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 series[edit]

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 rings[edit]

The unit group of the ring Mn(R) of n × n matrices over a commutative ring R (for example, a field) is the group GLn(R) of invertible matrices.

An element of the matrix ring is invertible if and only if the determinant of the element is invertible in R, with the inverse explicitly given by Cramer's rule.

In general[edit]

Let be a ring. For any in , if is invertible, then is invertible with the inverse .[5] The formula for the inverse can be found as follows: thinking formally, suppose is invertible and that the inverse is given by a geometric series: . Then, manipulating it formally,

See also Hua's identity for a similar type of results.

Group of units[edit]

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.


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


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).

See also[edit]


  1. ^ In a ring, the additive inverse of a non-zero element can equal the element itself.


  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.


  • Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9.
  • Jacobson, Nathan (2009). Basic Algebra 1 (2nd ed.). Dover. ISBN 978-0-486-47189-1.
  • Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Springer. ISBN 0-387-95385-X.
  • Watkins, John J. (2007), Topics in commutative ring theory, Princeton University Press, ISBN 978-0-691-12748-4, MR 2330411