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In mathematics, the notion of a divisor originally arose within the context of arithmetic of whole numbers. See the article on divisors for this simplest example. With the development of abstract rings, of which the integers are the archetype, the original notion of divisor found a natural extension. Divisibility is a useful concept for the analysis of the structure of commutative rings because of its relationship with the ideal structure of such rings. ==Definition== Let ''R'' be a ring,〔In this article, rings are assumed to have a 1.〕 and let ''a'' and ''b'' be elements of ''R''. If there exists an element ''x'' in ''R'' with , one says that ''a'' is a left divisor of ''b'' in ''R'' and that ''b'' is a right multiple of ''a''.〔Bourbaki, p. 97〕 Similarly, if there exists an element ''y'' in ''R'' with , one says that ''a'' is a right divisor of ''b'' and that ''b'' is a left multiple of ''a''. One says that ''a'' is a two-sided divisor of ''b'' if it is both a left divisor and a right divisor of ''b''. When ''R'' is commutative, a left divisor, a right divisor and a two-sided divisor coincide, so in this context one says that ''a'' is a divisor of ''b'', or that ''b'' is a multiple of ''a'', and one writes . Elements ''a'' and ''b'' of an integral domain are associates if both and . The associate relationship is an equivalence relation on ''R'', and hence divides ''R'' into disjoint equivalence classes. Notes: These definitions make sense in any magma ''R'', but they are used primarily when this magma is the multiplicative monoid of a ring. 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「Divisibility (ring theory)」の詳細全文を読む スポンサード リンク
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