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In abstract algebra, the length of a module is a measure of the module's "size". It is defined to be the length of the longest chain of submodules and is a generalization of the concept of dimension for vector spaces. Modules with ''finite'' length share many important properties with finite-dimensional vector spaces. Other concepts used to 'count' in ring and module theory are depth and height; these are both somewhat more subtle to define. There are also various ideas of ''dimension'' that are useful. Finite length commutative rings play an essential role in functorial treatments of formal algebraic geometry. == Definition == Let ''M'' be a (left or right) module over some ring ''R''. Given a chain of submodules of ''M'' of the form : we say that ''n'' is the ''length'' of the chain. The length of ''M'' is defined to be the largest length of any of its chains. If no such largest length exists, we say that ''M'' has infinite length. A ring ''R'' is said to have finite length as a ring if it has finite length as left ''R'' 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「length of a module」の詳細全文を読む スポンサード リンク
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