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In mathematics, and more specifically in algebra, a domain is a nonzero ring in which implies or .〔Lam (2001), p.3.〕 (Sometimes such a ring is said to "have the zero-product property.") Equivalently, a domain is a ring in which 0 is the only left zero divisor (or equivalently, the only right zero divisor). A commutative domain is called an integral domain.〔〔Rowen (1994), p. 99.〕 (Warning: The mathematical literature contains some variants of the definition of "domain".)〔Some authors also consider the zero ring to be a domain: see Polcino M. & Sehgal (2002), p. 65. Some authors apply the term "domain" also to rngs with the zero-product property; such authors consider ''n''Z to be a domain for each positive integer ''n'': see Lanski (2005), p. 343. But integral domains are always required to be nonzero and to have a 1.〕 == Examples and non-examples == * The ring Z/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 integer ''n'', the ring Z/''n''Z is a domain if and only if ''n'' is prime. * A ''finite'' domain is automatically a finite field, by Wedderburn's little theorem. * The quaternions form a noncommutative domain. More generally, any division algebra is a domain, since all its nonzero elements are invertible. * The set of all integral quaternions is a noncommutative ring which is a subring of quaternions, hence a noncommutative domain. * A matrix ring M''n''(''R'') 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 unit ''E''12 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 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. By the theorem below, the Weyl algebra is a domain. * The universal enveloping algebra of any Lie algebra over a field is a domain. The proof uses the standard filtration on the universal enveloping algebra and the Poincaré–Birkhoff–Witt theorem. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Domain (ring theory)」の詳細全文を読む スポンサード リンク
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