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In mathematics, and specifically in abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero.〔Bourbaki, p. 116.〕〔Dummit and Foote, p. 228.〕 Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain the cancellation property holds for multiplication by a nonzero element ''a'', that is, if , an equality implies . "Integral domain" is defined almost universally as above, but there is some variation. This article follows the convention that rings have a multiplicative identity, generally denoted 1, but some authors do not follow this, by not requiring integral domains to have a multiplicative identity.〔B.L. van der Waerden, Algebra Erster Teil, p. 36, Springer-Verlag, Berlin, Heidelberg 1966.〕〔I.N. Herstein, Topics in Algebra, p. 88-90, Blaisdell Publishing Company, London 1964.〕 Noncommutative integral domains are sometimes admitted.〔J.C. McConnel and J.C. Robson "Noncommutative Noetherian Rings" (Graduate Studies in Mathematics Vol. 30, AMS)〕 This article, however, follows the much more usual convention of reserving the term "integral domain" for the commutative case and using "domain" for the general case including noncommutative rings. Some sources, notably Lang, use the term entire ring for integral domain.〔Pages 91–92 of 〕 Some specific kinds of integral domains are given with the following chain of class inclusions: == Definitions == There are a number of equivalent definitions of integral domain: * An integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. * An integral domain is a nonzero commutative ring with no nonzero zero divisors. * An integral domain is a commutative ring in which the zero ideal is a prime ideal. * An integral domain is a nonzero commutative ring for which every non-zero element is cancellable under multiplication. * An integral domain is a ring for which the set of nonzero elements is a commutative monoid under multiplication (because the monoid is closed under multiplication). * An integral domain is a ring that is (isomorphic to) a subring of a field. (This implies it is a nonzero commutative ring.) * An integral domain is a nonzero commutative ring in which for every nonzero element ''r'', the function that maps each element ''x'' of the ring to the product ''xr'' is injective. Elements ''r'' with this property are called ''regular'', so it is equivalent to require that every nonzero element of the ring be regular. 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「integral domain」の詳細全文を読む スポンサード リンク
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