Domain (ring Theory) - Examples

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 algebra 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 ring of order greater than one is never a domain, since it has zero divisors, and even nilpotent elements. For example, the square of the matrix unit E₁₂ is zero.
• The tensor algebra of a vector space, 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 extension 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 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.