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- Domain_(ring_theory) abstract "In mathematics, and more specifically in algebra, 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[x, y]/(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.".
- Domain_(ring_theory) wikiPageID "487627".
- Domain_(ring_theory) wikiPageRevisionID "577281074".
- Domain_(ring_theory) hasPhotoCollection Domain_(ring_theory).
- Domain_(ring_theory) subject Category:Algebraic_structures.
- Domain_(ring_theory) subject Category:Ring_theory.
- Domain_(ring_theory) comment "In mathematics, and more specifically in algebra, 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).".
- Domain_(ring_theory) label "Anneau sans diviseur de zéro".
- Domain_(ring_theory) label "Domain (ring theory)".
- Domain_(ring_theory) label "Dominio (álgebra)".
- Domain_(ring_theory) label "非可換整域".
- Domain_(ring_theory) sameAs Dominio_(álgebra).
- Domain_(ring_theory) sameAs Anneau_sans_diviseur_de_zéro.
- Domain_(ring_theory) sameAs 非可換整域.
- Domain_(ring_theory) sameAs m.02gfdq.
- Domain_(ring_theory) sameAs Q2851442.
- Domain_(ring_theory) sameAs Q2851442.
- Domain_(ring_theory) wasDerivedFrom Domain_(ring_theory)?oldid=577281074.
- Domain_(ring_theory) isPrimaryTopicOf Domain_(ring_theory).