Domain (ring theory)

Last updated June 02, 2024

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]

Examples and non-examples

The ring $\mathbb {Z} /6\mathbb {Z}$ 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 $\mathbb {Z} /n\mathbb {Z}$ 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 ring is a domain, since every nonzero element is invertible.
The set of all Lipschitz quaternions, that is, quaternions of the form $a+bi+cj+dk$ where a, b, c, d are integers, is a noncommutative subring of the quaternions, hence a noncommutative domain.
Similarly, the set of all Hurwitz quaternions, that is, quaternions of the form $a+bi+cj+dk$ where a, b, c, d are either all integers or all half-integers, is 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.
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

(1-g)(1+g+\cdots +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.

Related Research Articles

In mathematics, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain, every nonzero element a has the cancellation property, that is, if a ≠ 0, an equality ab = ac implies b = c.

In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with the zero ideal.

In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field of rational numbers. Intuitively, it consists of ratios between integral domain elements.

In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal amongst all proper ideals. In other words, I is a maximal ideal of a ring R if there are no other ideals contained between I and R.

In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. Informally, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.

In abstract algebra, an element $a$ of a ring $R$ is called a left zero divisor if there exists a nonzero $x$ in $R$ such that $ax = 0$ , or equivalently if the map from $R$ to $R$ that sends $x$ to $ax$ is not injective. Similarly, an element $a$ of a ring is called a right zero divisor if there exists a nonzero $y$ in $R$ such that $ya = 0$ . This is a partial case of divisibility in rings. An element that is a left or a right zero divisor is simply called a zero divisor. An element $a$ that is both a left and a right zero divisor is called a two-sided zero divisor. If the ring is commutative, then the left and right zero divisors are the same.

In mathematics, a unique factorization domain (UFD) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is an integral domain in which every non-zero non-unit element can be written as a product of irreducible elements, uniquely up to order and units.

In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings.

In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily unique up to the order of the factors. There are at least three other characterizations of Dedekind domains that are sometimes taken as the definition: see below.

In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted for the possible factors, that is, the ring to which the coefficients of the polynomial and its possible factors are supposed to belong. For example, the polynomial $x 2 - 2$ is a polynomial with integer coefficients, but, as every integer is also a real number, it is also a polynomial with real coefficients. It is irreducible if it is considered as a polynomial with integer coefficients, but it factors as $if it is considered as a polynomial with real coefficients. One says that the polynomial x 2 - 2 is irreducible over the integers but not over the reals.$

In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the given group. As a ring, its addition law is that of the free module and its multiplication extends "by linearity" the given group law on the basis. Less formally, a group ring is a generalization of a given group, by attaching to each element of the group a "weighting factor" from a given ring.

In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates with coefficients in another ring, often a field.

In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral domain are like ideals where denominators are allowed. In contexts where fractional ideals and ordinary ring ideals are both under discussion, the latter are sometimes termed integral ideals for clarity.

In mathematics, a Hurwitz quaternion is a quaternion whose components are either all integers or all half-integers. The set of all Hurwitz quaternions is

In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal.

In mathematics, the characteristic of a ring $R$ , often denoted $char(R)$ , is defined to be the smallest positive number of copies of the ring's multiplicative identity ( $1$ ) that will sum to the additive identity ( $0$ ). If no such number exists, the ring is said to have characteristic zero.

In mathematics, specifically in ring theory, a torsion element is an element of a module that yields zero when multiplied by some non-zero-divisor of the ring. The torsion submodule of a module is the submodule formed by the torsion elements. A torsion module is a module consisting entirely of torsion elements. A module is torsion-free if its only torsion element is the zero element.

In algebra, the zero-product property states that the product of two nonzero elements is nonzero. In other words,

In mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist a and b in the ring such that ab and ba are different. Equivalently, a noncommutative ring is a ring that is not a commutative ring.

This is a glossary of algebraic geometry.

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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[Lam-1] 1 2 Lam (2001), p. 3

[2] Rowen (1994), p. 99.

[3] 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.

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