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In linear algebra, the Dieudonné determinant is a generalization of the determinant of a matrix to matrices over division rings and local rings. It was introduced by . If ''K'' is a division ring, then the Dieudonné determinant is a homomorphism of groups from the group GL''n''(''K'') of invertible ''n'' by ''n'' matrices over ''K'' onto the abelianization ''K'' */(''K'' * ) of the multiplicative group ''K'' * of ''K''. For example, the Dieudonné determinant for a 2-by-2 matrix is : ==Properties== Let ''R'' be a local ring. There is a determinant map from the matrix ring GL(''R'') to the abelianised unit group ''R''∗ab with the following properties:〔Rosenberg (1994) p.64〕 * The determinant is invariant under elementary row operations * The determinant of the identity is 1 * If a row is left multiplied by ''a'' in ''R''∗ then the determinant is left multiplied by ''a'' * The determinant is multiplicative: det(''AB'') = det(''A'')det(''B'') * If two rows are exchanged, the determinant is multiplied by −1 * The determinant is invariant under transposition 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「Dieudonné determinant」の詳細全文を読む スポンサード リンク
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