Domain (ring theory)

Domain (ring theory)

In mathematics, especially in the area of abstract algebra known as ring theory, a domain is a ring such that ab = 0 implies that either a = 0 or b = 0.[1] That is, it is a ring which has no left or right zero divisors. (Sometimes such a ring is said to "have the zero-product property.") Some authors require the ring to be nontrivial (that is, it must have more than one element).[2] If the domain has a multiplicative identity (which we may call 1), this is equivalent to saying that 1 ≠ 0[3] Thus a domain is a nontrivial ring without left or right zero divisors. A commutative domain with 1 ≠ 0 is called an integral domain.[4]

A finite domain is automatically a finite field by Wedderburn's little theorem.

Zero divisors have a topological interpretation, at least in the case of commutative rings: a ring R is an integral domain, if and only if it is reduced and its spectrum Spec R is an irreducible topological space. The first property is often considered to encode some infinitesimal information, whereas the second one is more geometric.

An example: the ring k[x, y]/(xy), where k is a field, is not a domain, as the images of x and y in this ring are zero divisors. Geometrically, this corresponds to the fact that the spectrum of this ring, which is the union of the lines x = 0 and y = 0, is not irreducible. Indeed, these two lines are its irreducible components.


Constructions of domains

One way of proving that a ring is a domain is by exhibiting a filtration with special properties.

Theorem: If R is a filtered ring whose associated graded ring gr R is a domain, then R itself is a domain.

This theorem needs to be complemented by the analysis of the graded ring gr R.


  • The ring nZ is a domain (for each integer n > 1) but not an integral domain since 1 \not \in n\mathbb{Z}.[5]
  • The quaternions form a noncommutative domain. More generally, any division algebra is a domain, since all its non-zero elements are invertible.
  • The set of all integral quaternions is a noncommutative ring which is a subring of quaternions, hence a noncommutative domain.
  • The matrix ring of order greater than one is never a domain, since it has zero divisors, and even nilpotent elements. For example, the square of the matrix unit E12 is zero.
  • The tensor algebra of a vector space, or equivalently, the algebra of polynomials in noncommuting variables over a field,  \mathbb{K}\langle x_1,\ldots,x_n\rangle, is a domain. This may be proved using an ordering on the noncommutative monomials.
  • If R is a domain and S is an Ore extension of R then S is a domain.
  • The Weyl algebra is a noncommutative domain. Indeed, it has two natural filtrations, by the degree of the derivative and by the total degree, and the associated graded ring for either one is isomorphic to the ring of polynomials in two variables. By the theorem above, the Weyl algebra is a domain.
  • The universal enveloping algebra of any Lie algebra over a field is a domain. The proof uses the standard filtration on the universal enveloping algebra and the Poincaré–Birkhoff–Witt theorem.

Group rings and the zero divisor problem

Suppose that G is a group and K is a field. Is the group ring R = K[G] a domain? The identity


shows that an element g of finite order n induces a zero divisor 1−g in R. The zero divisor problem asks whether this is the only obstruction, in other words,

Given a field K and a torsion-free group G, is it true that K[G] contains no zero divisors?

No countexamples are known, but the problem remains open in general (as of 2007).

For many special classes of groups, the answer is affirmative. Farkas and Snider proved in 1976 that if G is a torsion-free polycyclic-by-finite group and char K = 0 then the group ring K[G] is a domain. Later (1980) Cliff removed the restriction on the characteristic of the field. In 1988, Kropholler, Linnell and Moody generalized these results to the case of torsion-free solvable and solvable-by-finite groups. Earlier (1965) work of Lazard, whose importance was not appreciated by the specialists in the field for about 20 years, had dealt with the case where K is the ring of p-adic integers and G is the pth congruence subgroup of GL(n,Z).

See also


  1. ^ Polcino M. & Sehgal (2002), p. 65.
  2. ^ Lanski (2005), p. 343, Definition 10.18.
  3. ^ Jacobson (2009), p. 90, Section 2.2. Note that if 1=0, then a=1a=0a=0 showing that all elements are 0.
  4. ^ Rowen (1994), p. 99.
  5. ^ Lanski (2005), p. 343, Definition 10.18.


  • Lam, Tsit-Yuen (2001). A First Course in Noncommutative Rings (2nd ed.). Berlin, New York: Springer-Verlag. ISBN 978-0-387-95325-0. MR1838439. 
  • Charles Lanski (2005). Concepts in abstract algebra. AMS Bookstore. ISBN 053442323X. 
  • César Polcino Milies; Sudarshan K. Sehgal (2002). An introduction to group rings. Springer. ISBN 1402002386. 
  • Nathan Jacobson (2009). Basic Algebra I. Dover. ISBN 978-0-486-47189-1. 
  • Louis Halle Rowen (1994). Algebra: groups, rings, and fields. A K Peters. ISBN 1568810288. 

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Ring theory — In abstract algebra, ring theory is the study of rings algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers. Ring theory studies the structure of rings, their… …   Wikipedia

  • Ideal (ring theory) — In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like even number or multiple of 3 . For instance, in… …   Wikipedia

  • Height (ring theory) — In commutative algebra, the height of an prime ideal mathfrak{p} in a ring R is the number of strict inclusions in the longest chain of prime ideals contained in mathfrak{p} [Matsumura,Hideyuki: Commutative Ring Theory ,page 30 31,1989 ] . Then… …   Wikipedia

  • Divisibility (ring theory) — In mathematics, the notion of a divisor originally arose within the context of arithmetic of whole numbers. Please see the page about divisors for this simplest example. With the development of abstract rings, of which the integers are the… …   Wikipedia

  • Unit (ring theory) — In mathematics, a unit in a (unital) ring R is an invertible element of R , i.e. an element u such that there is a v in R with : uv = vu = 1 R , where 1 R is the multiplicative identity element.That is, u is an invertible element of the… …   Wikipedia

  • Completion (ring theory) — In abstract algebra, a completion is any of several related functors on rings and modules that result in complete topological rings and modules. Completion is similar to localization, and together they are among the most basic tools in analysing… …   Wikipedia

  • Glossary of ring theory — Ring theory is the branch of mathematics in which rings are studied: that is, structures supporting both an addition and a multiplication operation. This is a glossary of some terms of the subject. Contents 1 Definition of a ring 2 Types of… …   Wikipedia

  • Domain — may refer to: General Territory (administrative division), a non sovereign geographic area which has come under the authority of another government Public domain, a body of works and knowledge without proprietary interest Eminent domain, the… …   Wikipedia

  • Ring (mathematics) — This article is about algebraic structures. For geometric rings, see Annulus (mathematics). For the set theory concept, see Ring of sets. Polynomials, represented here by curves, form a ring under addition and multiplication. In mathematics, a… …   Wikipedia

  • Ring homomorphism — In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication. More precisely, if R and S are rings, then a ring homomorphism is a function f : R → S such that …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.