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In the context of a module ''M'' over a ring ''R'', the top of ''M'' is the largest semisimple quotient module of ''M'' if it exists. For finite-dimensional ''k''-algebras (''k'' a field), if rad(''M'') denotes the intersection of all proper maximal submodules of ''M'' (the radical of the module), then the top of ''M'' is ''M''/rad(''M''). In the case of local rings with maximal ideal ''P'', the top of ''M'' is ''M''/''PM''. In general if ''R'' is a semilocal ring (=semi-artinian ring), that is, if ''R''/Rad(''R'') is an Artinian ring, where Rad(''R'') is the Jacobson radical of ''R'', then ''M''/rad(''M'') is a semisimple module and is the top of ''M''. This includes the cases of local rings and finite dimensional algebras over fields. ==See also== *Projective cover *Radical of a module *Socle (mathematics) 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「Top (mathematics)」の詳細全文を読む スポンサード リンク
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