Domain (ring theory)

In mathematics, and more specifically in algebra, a domain is a nonzero ring in which ab = 0 implies a = 0 or b = 0.^[1] (Sometimes such a ring is said to "have the zero-product property.") Equivalently, a domain is a ring in which 0 is the only left zero divisor (or equivalently, the only right zero divisor). A commutative domain is called an integral domain.^[1]^[2] Mathematical literature contains multiple variants of the definition of "domain".^[3]

Algebraic structures
Group-like Semigroup / Monoid Racks and quandles Quasigroup and loop Abelian group Magma Lie group Group theory
Ring-like Ring Semiring Near-ring Commutative ring Integral domain Field Division ring Ring theory
Lattice-like Lattice Semilattice Map of lattices Lattice theory
Module-like Module Group with operators Vector space Linear algebra
Algebra-like Algebra Associative Non-associative Composition algebra Lie algebra Graded Bialgebra

Examples and non-examples

The ring Z/6Z is not a domain, because the images of 2 and 3 in this ring are nonzero elements with product 0. More generally, for a positive integer n, the ring Z/nZ is a domain if and only if n is prime.
A finite domain is automatically a finite field, by Wedderburn's little theorem.
The quaternions form a noncommutative domain. More generally, any division algebra is a domain, since all its nonzero elements are invertible.
The set of all integral quaternions is a noncommutative ring which is a subring of quaternions, hence a noncommutative domain.
A matrix ring M_n(R) for n ≥ 2 is never a domain: if R is nonzero, such a matrix ring has nonzero zero divisors and even nilpotent elements other than 0. For example, the square of the matrix unit E₁₂ is 0.
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 below, 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.

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).

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

(1-g)(1+g+\ldots +g^{{n-1}})=1-g^{n},

shows that an element g of finite order n > 1 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 counterexamples are known, but the problem remains open in general (as of 2017).

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 Michel 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).

Spectrum of an integral domain

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, since 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.

Notes

1 2 Lam (2001), p.3.
↑ Rowen (1994), p. 99.
↑ Some authors also consider the zero ring to be a domain: see Polcino M. & Sehgal (2002), p. 65. Some authors apply the term "domain" also to rngs with the zero-product property; such authors consider nZ to be a domain for each positive integer n: see Lanski (2005), p. 343. But integral domains are always required to be nonzero and to have a 1.

References

Lam, Tsit-Yuen (2001). A First Course in Noncommutative Rings (2nd ed.). Berlin, New York: Springer-Verlag. ISBN 978-0-387-95325-0. MR 1838439.
Charles Lanski (2005). Concepts in abstract algebra. AMS Bookstore. ISBN 0-534-42323-X.
César Polcino Milies; Sudarshan K. Sehgal (2002). An introduction to group rings. Springer. ISBN 1-4020-0238-6.
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 1-56881-028-8.

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