Inalgebra, adomain is anonzeroring in whichab = 0 impliesa = 0 orb = 0.^[1] (Sometimes such a ring is said to "have thezero-product property".) Equivalently, a domain is a ring in which 0 is the only leftzero divisor (or equivalently, the only right zero divisor). Acommutative domain is called anintegral domain.^[1][2] Mathematical literature contains multiple variants of the definition of "domain".^[3]

• The ring${\displaystyle \mathbb{Z}/6\mathbb{Z}}$ 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${\displaystyle n}$, the ring${\displaystyle \mathbb{Z}/n\mathbb{Z}}$ is a domain if and only if${\displaystyle n}$ is prime.
• Afinite domain is automatically afinite field, byWedderburn's little theorem.
• Thequaternions form a noncommutative domain. More generally, anydivision ring is a domain, since every nonzero element isinvertible.
• The set of allLipschitz quaternions, that is, quaternions of the form${\displaystyle a+bi+cj+dk}$ wherea,b,c,d are integers, is a noncommutative subring of the quaternions, hence a noncommutative domain.
• Similarly, the set of allHurwitz quaternions, that is, quaternions of the form${\displaystyle a+bi+cj+dk}$ wherea,b,c,d are either all integers or allhalf-integers, is a noncommutative domain.
• Amatrix ring M_n(R) forn ≥ 2 is never a domain: ifR is nonzero, such a matrix ring has nonzero zero divisors and evennilpotent elements other than 0. For example, the square of thematrix unitE₁₂ is 0.
• Thetensor algebra of avector space, or equivalently, the algebra of polynomials in noncommuting variables over a field,${\displaystyle \mathbb{K}⟨x_1,\dots ,x_n⟩,}$ is a domain. This may be proved using an ordering on the noncommutative monomials.
• IfR is a domain andS is anOre extension ofR thenS is a domain.
• TheWeyl algebra is a noncommutative domain.
• Theuniversal enveloping algebra of anyLie algebra over a field is a domain. The proof uses the standard filtration on the universal enveloping algebra and thePoincaré–Birkhoff–Witt theorem.

Suppose thatG is agroup andK is afield. Is thegroup ringR =K[G] a domain? The identity

${\displaystyle (1-g)(1+g+\cdots +g^{n-1})=1-g^n,}$

shows that an elementg of finiteordern > 1 induces a zero divisor1 −g inR. Thezero divisor problem asks whether this is the only obstruction; in other words,

Given afieldK and atorsion-free groupG, is it true thatK[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 ifG is a torsion-freepolycyclic-by-finite group andcharK = 0 then the group ringK[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-freesolvable and solvable-by-finite groups. Earlier (1965) work ofMichel Lazard, whose importance was not appreciated by the specialists in the field for about 20 years, had dealt with the case whereK is the ring ofp-adic integers andG is thepthcongruence subgroup ofGL(n,Z).

Zero divisors have a topological interpretation, at least in the case of commutative rings: a ringR is an integral domain if and only if it isreduced and itsspectrum SpecR is anirreducible topological space. The first property is often considered to encode some infinitesimal information, whereas the second one is more geometric.

An example: the ringk[x,y]/(xy), wherek is a field, is not a domain, since the images ofx andy in this ring are zero divisors. Geometrically, this corresponds to the fact that the spectrum of this ring, which is the union of the linesx = 0 andy = 0, is not irreducible. Indeed, these two lines are its irreducible components.

• Zero divisor
• Zero-product property
• Divisor (ring theory)
• Integral domain

• 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.

• Ring theory
• Algebraic structures

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