Editing Domain (ring theory) (section)

== 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]].