Domain (ring Theory)
In mathematics, especially in the area of abstract algebra known as ring theory, a domain is a ring such that ab = 0 implies that either a = 0 or b = 0. That is, it is a ring which has no left or right zero divisors. (Sometimes such a ring is said to "have the zero-product property.") Some authors require the ring to be nontrivial (that is, it must have more than one element). If the domain has a multiplicative identity (which we may call 1), this is equivalent to saying that 1 ≠ 0 Thus a domain is a nontrivial ring without left or right zero divisors. A commutative domain with 1 ≠ 0 is called an integral domain.
A finite domain is automatically a finite field by Wedderburn's little theorem.
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 k/(xy), 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.
Read more about Domain (ring Theory): Constructions of Domains, Examples, Group Rings and The Zero Divisor Problem
Famous quotes containing the word domain:
“Without metaphor the handling of general concepts such as culture and civilization becomes impossible, and that of disease and disorder is the obvious one for the case in point. Is not crisis itself a concept we owe to Hippocrates? In the social and cultural domain no metaphor is more apt than the pathological one.”
—Johan Huizinga (1872–1945)