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In mathematics, specifically module theory, the annihilator of a set is a concept generalizing torsion and orthogonality. ==Definitions== Let ''R'' be a ring, and let ''M'' be a left ''R''-module. Choose a nonempty subset ''S'' of ''M''. The annihilator, denoted Ann''R''(''S''), of ''S'' is the set of all elements ''r'' in ''R'' such that for each ''s'' in ''S'', .〔Pierce (1982), p. 23.〕 In set notation, : It is the set of all elements of ''R'' that "annihilate" ''S'' (the elements for which ''S'' is torsion). Subsets of right modules may be used as well, after the modification of "" in the definition. The annihilator of a single element ''x'' is usually written Ann''R''(''x'') instead of Ann''R''(). If the ring ''R'' can be understood from the context, the subscript ''R'' can be omitted. Since ''R'' is a module over itself, ''S'' may be taken to be a subset of ''R'' itself, and since ''R'' is both a right and a left ''R'' module, the notation must be modified slightly to indicate the left or right side. Usually and or some similar subscript scheme are used to distinguish the left and right annihilators, if necessary. If ''M'' is an ''R''-module and , then ''M'' is called a faithful module. 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「Annihilator (ring theory)」の詳細全文を読む スポンサード リンク
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