Domain (ring Theory)
   HOME

TheInfoList



OR:

In
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
, a domain is a nonzero
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
in which implies or .Lam (2001), p. 3 (Sometimes such a ring is said to "have the
zero-product property In algebra, the zero-product property states that the product of two nonzero elements is nonzero. In other words, \textab=0,\texta=0\textb=0. This property is also known as the rule of zero product, the null factor law, the multiplication proper ...
".) Equivalently, a domain is a ring in which 0 is the only left
zero divisor In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zero ...
(or equivalently, the only right zero divisor). A
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
domain is called an
integral domain In mathematics, specifically abstract algebra, 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 set ...
. Mathematical literature contains multiple variants of the definition of "domain".Some authors also consider the
zero ring In ring theory, a branch of mathematics, the zero ring or trivial ring is the unique ring (up to isomorphism) consisting of one element. (Less commonly, the term "zero ring" is used to refer to any rng of square zero, i.e., a rng in which for a ...
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 ''n''Z 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.


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/''n''Z is a domain if and only if ''n'' is prime. * A ''finite'' domain is automatically a
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
, by
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, division rings and fields. The Artin–Zorn theorem generalizes the theorem to al ...
. * The
quaternions In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quater ...
form a noncommutative domain. More generally, any
division algebra In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible. Definitions Formally, we start with a non-zero algebra ''D'' over a fie ...
is a domain, since all its nonzero elements are
invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
. * The set of all integral quaternions is a noncommutative ring which is a subring of quaternions, hence a noncommutative domain. * A
matrix ring In abstract algebra, a matrix ring is a set of matrices with entries in a ring ''R'' that form a ring under matrix addition and matrix multiplication . The set of all matrices with entries in ''R'' is a matrix ring denoted M''n''(''R'')Lang, ''U ...
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 In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the class ...
elements other than 0. For example, the square of the matrix unit ''E''12 is 0. * The
tensor algebra In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', in the sense of being ...
of a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
, or equivalently, the algebra of polynomials in noncommuting variables over a field, \mathbb\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 In mathematics, especially in the area of algebra known as ring theory, an Ore extension, named after Øystein Ore, is a special type of a ring extension whose properties are relatively well understood. Elements of a Ore extension are called Ore pol ...
of ''R'' then ''S'' is a domain. * The
Weyl algebra In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable), namely expressions of the form : f_m(X) \partial_X^m + f_(X) \partial_X^ + \cdots + f_1(X) \partial_X + f_0(X). More prec ...
is a noncommutative domain. * The
universal enveloping algebra In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal enveloping algebras are used in the representati ...
of any
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
over a field is a domain. The proof uses the standard filtration on the universal enveloping algebra and the
Poincaré–Birkhoff–Witt theorem In mathematics, more specifically in the theory of Lie algebras, the Poincaré–Birkhoff–Witt theorem (or PBW theorem) is a result giving an explicit description of the universal enveloping algebra of a Lie algebra. It is named after Henri Poi ...
.


Group rings and the zero divisor problem

Suppose that ''G'' is a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
and ''K'' is a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
. Is the
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 the set of elements of the give ...
a domain? The identity : (1-g)(1+g+\cdots+g^)=1-g^n, shows that an element ''g'' of finite order induces a zero divisor in ''R''. The zero divisor problem asks whether this is the only obstruction; in other words, : Given a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
''K'' and a
torsion-free group 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 ...
''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 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 Michel Paul Lazard (5 December 1924 – 15 September 1987) was a French mathematician who worked on the theory of Lie groups in the context of p-adic analysis. Career and research Born in Paris, Lazard studied at the University of Paris–Sorbon ...
, 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 In mathematics, the -adic number system for any prime number  extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension ...
and ''G'' is the ''p''th
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 2 × 2 integer matrices of determinant 1, in which the ...
of .


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 A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors i ...
Spec ''R'' is an
irreducible topological space In the mathematical field of topology, a hyperconnected space or irreducible space is a topological space ''X'' that cannot be written as the union of two proper closed sets (whether disjoint or non-disjoint). The name ''irreducible space'' is pre ...
. The first property is often considered to encode some infinitesimal information, whereas the second one is more geometric. An example: the ring , 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 and , is not irreducible. Indeed, these two lines are its irreducible components.


See also

*
Zero divisor In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zero ...
*
Zero-product property In algebra, the zero-product property states that the product of two nonzero elements is nonzero. In other words, \textab=0,\texta=0\textb=0. This property is also known as the rule of zero product, the null factor law, the multiplication proper ...
* Divisor (ring theory) *
Integral domain In mathematics, specifically abstract algebra, 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 set ...


Notes


References

* * * * * {{DEFAULTSORT:Domain (Ring Theory) Ring theory Algebraic structures