Domain (ring theory)

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In abstract algebra, a domain is a ring with 0 ≠ 1 such that ab = 0 implies that either a = 0 or b = 0 (the zero-product property). That is, it is a nontrivial ring without left or right zero divisors.

A commutative domain is called an integral domain.

Zero-divisors have a geometric 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, where the second one is of geometric nature.

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.

1 Constructions of domains
2 Examples
3 Zero divisor problem
4 See also

[edit] 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.

[edit] Examples

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.

[edit] 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 is a zero divisor 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).