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In abstract algebra, a multiplicatively closed set (or multiplicative set) is a subset ''S'' of a ring ''R'' such that the following two conditions hold:〔Atiyah and Macdonald, p. 36.〕〔Lang, p. 107.〕 * . * For all ''x'' and ''y'' in ''S'', the product ''xy'' is in ''S''. In other words, ''S'' is closed under taking finite products, including the empty product 1.〔Eisenbud, p. 59.〕 Equivalently, a multiplicative set is a submonoid of the multiplicative monoid of a ring. Multiplicative sets are important especially in commutative algebra, where they are used to build localizations of commutative rings. A subset ''S'' of a ring ''R'' is called saturated if it is closed under taking divisors: i.e., whenever a product ''xy'' is in ''S'', the elements ''x'' and ''y'' are in ''S'' too. ==Examples== Common examples of multiplicative sets include: * the set-theoretic complement of a prime ideal in a commutative ring; * the set , where ''x'' is a fixed element of the ring; * the set of units of the ring; * the set of non-zero-divisors of the ring; * 1 + ''I'' for an ideal ''I''. 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「multiplicatively closed set」の詳細全文を読む スポンサード リンク
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