The ringZ/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 integern, the ringZ/nZ is a domain if and only ifn is prime.
The set of all integral quaternions is a noncommutative ring which is a subring of quaternions, hence a noncommutative domain.
A matrix ring Mn 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 unitE12 is 0.
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 is a domain by the [|theorem below], as 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.
One way of proving that a ring is a domain is by exhibiting a filtration with special properties. Theorem: If R is a filtered ring whose associated graded ring gr is a domain, then R itself is a domain. This theorem needs to be complemented by the analysis of the graded ring gr.
Suppose that G is a group and K is a field. Is the group ring a domain? The identity 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, No counterexamples are known, but the problem remains open in general. 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 is a domain. Later 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 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 pth congruence subgroup 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 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, 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.
