Domain (ring theory)
Encyclopedia
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. That is, it is a ring which has no left or right zero divisor
s. (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). If the domain has a multiplicative identity (which we may call 1), this is equivalent to saying that 1 ≠ 0 Thus a domain is a nontrivial ring without left or right zero divisors. A commutative
domain with 1 ≠ 0 is called an integral domain.
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.
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.
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,
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).
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, especially in the area of abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
known as ring theory
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...
, a domain is a ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
such that ab = 0 implies that either a = 0 or b = 0. That is, it is a ring which has no left or right zero divisor
Zero divisor
In abstract algebra, a nonzero element a of a ring is a left zero divisor if there exists a nonzero b such that ab = 0. Similarly, a nonzero element a of a ring is a right zero divisor if there exists a nonzero c such that ca = 0. An element that is both a left and a right zero divisor is simply...
s. (Sometimes such a ring is said to "have the zero-product property
Zero-product property
In the mathematical areas of algebra and analysis, the zero-product property, generally known as the nonexistence of zero divisors, and also called the zero-product rule, the rule of zero product, or any other similar name, is an abstract and explicit statement of the familiar property from...
.") Some authors require the ring to be nontrivial (that is, it must have more than one element). If the domain has a multiplicative identity (which we may call 1), this is equivalent to saying that 1 ≠ 0 Thus a domain is a nontrivial ring without left or right zero divisors. A commutative
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
domain with 1 ≠ 0 is called an integral domain.
A finite domain is automatically a finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
by Wedderburn's little theorem
Wedderburn's little theorem
In mathematics, Wedderburn's little theorem states that every finite domain is a field. In other words, for finite rings, there is no distinction between domains, skew-fields and fields.The Artin–Zorn theorem generalizes the theorem to alternative rings....
.
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
Reduced ring
In ring theory, a ring R is called a reduced ring if it has no non-zero nilpotent elements. Equivalently, a ring is reduced if it has no non-zero elements with square zero, that is, x2 = 0 implies x = 0...
and its spectrum
Spectrum of a ring
In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec, is the set of all proper prime ideals of R...
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
Filtered algebra
In mathematics, a filtered algebra is a generalization of the notion of a graded algebra. Examples appear in many branches of mathematics, especially in homological algebra and representation theory....
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
Graded algebra
In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a gradation ....
gr R.
Examples
- The ring nZ is a domain (for each integer n > 1) but not an integral domain since
. - The quaternions form a noncommutative domain. More generally, any division algebraDivision algebraIn the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field, in which division is possible.- Definitions :...
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 ringMatrix ringIn abstract algebra, a matrix ring is any collection of matrices forming a ring under matrix addition and matrix multiplication. The set of n×n matrices with entries from another ring is a matrix ring, as well as some subsets of infinite matrices which form infinite matrix rings...
of order greater than one is never a domain, since it has zero divisors, and even nilpotentNilpotentIn mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0....
elements. For example, the square of the matrix unitMatrix unitIn mathematics, a matrix unit is an idealisation of the concept of a matrix, with a focus on the algebraic properties of matrix multiplication. The topic is comparatively obscure within linear algebra, because it entirely ignores the numeric properties of matrices; it is mostly encountered in the...
E12 is zero. - The tensor algebraTensor algebraIn mathematics, the tensor algebra of a vector space V, denoted T or T•, is the algebra of tensors on V with multiplication being the tensor product...
of a vector spaceVector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
, or equivalently, the algebra of polynomials in noncommuting variables over a field,
is a domain. This may be proved using an ordering on the noncommutative monomials. - If R is a domain and S is an Ore extensionOre extensionIn mathematics, especially in the area of algebra known as ring theory, an Ore extension, named after Oystein Ore, is a special type of a ring extension whose properties are relatively well understood...
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 algebraUniversal enveloping algebraIn mathematics, for any Lie algebra L one can construct its universal enveloping algebra U. This construction passes from the non-associative structure L to a unital associative algebra which captures the important properties of L.Any associative algebra A over the field K becomes a Lie algebra...
of any Lie algebraLie algebraIn mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
over a field is a domain. The proof uses the standard filtration on the universal enveloping algebra and the Poincaré–Birkhoff–Witt theoremPoincaré–Birkhoff–Witt theoremIn the theory of Lie algebras, the Poincaré–Birkhoff–Witt theorem is a result giving an explicit description of the universal enveloping algebra of a Lie algebra...
.
Group rings and the zero divisor problem
Suppose that G is a groupGroup (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
and K is a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
. Is the group ring
Group ring
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 one-to-one with the given group. As a ring, its addition law is that of the free...
R = K[G] a domain? The identity
shows that an element g of finite order
Order (group theory)
In group theory, a branch of mathematics, the term order is used in two closely related senses:* The order of a group is its cardinality, i.e., the number of its elements....
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 fieldField (mathematics)In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
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
Polycyclic group
In mathematics, a polycyclic group is a solvable group that satisfies the maximal condition on subgroups...
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
Solvable group
In mathematics, more specifically in the field of group theory, a solvable group is a group that can be constructed from abelian groups using extensions...
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
Congruence subgroup
In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example would be invertible 2x2 integer matrices of determinant 1, such that the off-diagonal entries are even.An importance class of congruence...
of GL(n,Z).