Editing Domain (ring theory)

{{Short description|Unital ring with no zero divisors other than 0; noncommutative generalization of integral domains}}
In [[algebra]], a '''domain''' is a [[zero ring|nonzero]] [[ring (mathematics)|ring]] in which {{nowrap|1=''ab'' = 0}} implies {{nowrap|1=''a'' = 0}} or {{nowrap|1=''b'' = 0}}.<ref name="Lam">Lam (2001), p. 3</ref> (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 ring|commutative]] domain is called an [[integral domain]].<ref name="Lam" /><ref>Rowen (1994), p. 99.</ref>  Mathematical literature contains multiple variants of the definition of "domain".<ref>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 [[rng (mathematics)|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.</ref>

{{Algebraic structures |Ring}}

== Examples and non-examples ==

* The ring <math>\mathbb{Z}/6\mathbb{Z}</math> 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 <math>n</math>, the ring [[Modular_arithmetic#Integers_modulo_m|<math>\mathbb{Z}/n\mathbb{Z}</math>]] is a domain if and only if <math>n</math> 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 element|invertible]].
* The set of all [[Lipschitz quaternion]]s, that is, quaternions of the form <math>a+bi+cj+dk</math> where ''a'', ''b'', ''c'', ''d'' are integers, is a noncommutative subring of the quaternions, hence a noncommutative domain.
* Similarly, the set of all [[Hurwitz quaternion]]s, that is, quaternions of the form <math>a+bi+cj+dk</math> where ''a'', ''b'', ''c'', ''d'' are either all integers or all [[half-integers]], is a noncommutative domain.
* A [[matrix ring]] M<sub>''n''</sub>(''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]] ''E''<sub>12</sub> is 0.
* The [[tensor algebra]] of a [[vector space]], or equivalently, the algebra of polynomials in noncommuting variables over a field, <math> \mathbb{K}\langle x_1,\ldots,x_n\rangle, </math> 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 (mathematics)|group]] and ''K'' is a [[field (mathematics)|field]]. Is the [[group ring]] {{nowrap|1=''R'' = ''K''[''G'']}} a domain?  The identity

: <math> (1-g)(1+g+\cdots+g^{n-1})=1-g^n,</math>

shows that an element ''g'' of finite [[order (group theory)|order]] {{nowrap|''n'' > 1}} induces a zero divisor {{nowrap|1 − ''g''}} in ''R''. The '''zero divisor problem''' asks whether this is the only obstruction; in other words,

: Given a [[field (mathematics)|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 group|polycyclic-by-finite]] group and {{nowrap|1=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 group|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 ''p''th [[congruence subgroup]] of {{nowrap|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 ring|reduced]] and its [[Spectrum of a ring|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 {{nowrap|''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 {{nowrap|1=''x'' = 0}} and {{nowrap|1=''y'' = 0}}, is not irreducible. Indeed, these two lines are its irreducible components.

== See also ==
* [[Zero divisor]]
* [[Zero-product property]]
* [[Divisor (ring theory)]]
* [[Integral domain]]

== Notes ==
{{Reflist}}

== References ==
* {{Cite book | last1=Lam | first1=Tsit-Yuen | title=A First Course in Noncommutative Rings | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | isbn=978-0-387-95325-0 | mr=1838439 | year=2001|url=https://books.google.com/books?id=VtvwJzpWBqUC&q=domain}}
* {{cite book | author=Charles Lanski | title=Concepts in abstract algebra | publisher=AMS Bookstore | year=2005 | isbn=0-534-42323-X }}
* {{cite book | author=César Polcino Milies | author2=Sudarshan K. Sehgal | title=An introduction to group rings | publisher=Springer | year=2002 | isbn=1-4020-0238-6 |url=https://books.google.com/books?id=7m9P9hM4pCQC&q=domain}}
* {{cite book | author=Nathan Jacobson | author-link=Nathan Jacobson | title=Basic Algebra I | publisher=Dover | year=2009 | isbn=978-0-486-47189-1 }}
* {{cite book | author=Louis Halle Rowen | title=Algebra: groups, rings, and fields | publisher=[[A K Peters]] | year=1994 | isbn=1-56881-028-8 }}

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[[Category:Ring theory]]
[[Category:Algebraic structures]]