Domain (ring theory)
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In abstract algebra, a domain is a ring with 0 ≠ 1 such that ab = 0 implies that either a = 0 or b = 0. That is, it is a nontrivial ring without left or right zero divisors.
A commutative domain is called an integral domain.
Zero-divisors have a geometric 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 SpecR is an irreducible topological space. The first property is often considered to encode some infinitesimal information, where the second one is of geometric nature.
An example: the ring k[x,y] / (x * y), where k is a field, is not a domain, as 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 x = 0 and y = 0, is not irreducible. Indeed, these two lines are its irreducible components.
[edit] Examples
The quaternions are a non-commutative domain.
The Weyl algebra is the ring of differential operators with polynomial coefficients.
If R is a commutative ring and p ⊂ R is a prime ideal, then R / p is a domain.
A regular local ring is a domain.
