Editing Domain (ring theory) (section)

== 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 ring|reduced]] and its [[Spectrum of a ring|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 {{nowrap|''k''[''x'', ''y'']/(''xy'')}}, 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 {{nowrap|1=''x'' = 0}} and {{nowrap|1=''y'' = 0}}, is not irreducible. Indeed, these two lines are its irreducible components.