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In mathematics, more specifically in the field of ring theory, a ring has the invariant basis number (IBN) property if all finitely generated free left modules over ''R'' have a well-defined rank. In the case of fields, the IBN property becomes the statement that finite-dimensional vector spaces have a unique dimension. ==Definition== A ring ''R'' has invariant basis number (IBN) if for all positive integers ''m'' and ''n'', ''R''''m'' isomorphic to ''R''''n'' (as left ''R''-modules) implies that . Equivalently, this means there do not exist distinct positive integers ''m'' and ''n'' such that ''R''''m'' is isomorphic to ''R''''n''. Rephrasing the definition of invariant basis number in terms of matrices, it says that, whenever ''A'' is an ''m''-by-''n'' matrix over ''R'' and ''B'' is an ''n''-by-''m'' matrix over ''R'' such that and , then . This form reveals that the definition is left-right symmetric, so it makes no difference whether we define IBN in terms of left or right modules; the two definitions are equivalent. Note that the isomorphisms in the definitions are ''not'' ring isomorphisms, they are module isomorphisms. 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「Invariant basis number」の詳細全文を読む スポンサード リンク
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