|
In algebra, an Artin algebra is an algebra Λ over a commutative Artin ring ''R'' that is a finitely generated ''R''-module. They are named after Emil Artin. Every Artin algebra is an Artin ring. ==Dual and transpose== There are several different dualities taking finitely generated modules over Λ to modules over the opposite algebra Λop. *If ''M'' is a left Λ module then the right Λ-module ''M'' * is defined to be HomΛ(''M'',Λ). * The dual ''D''(''M'') of a left Λ-module ''M'' is the right Λ-module ''D''(''M'') = Hom''R''(''M'',''J''), where ''J'' is the dualizing module of ''R'', equal to the sum of the injective envelopes of the non-isomorphic simple ''R''-modules or equivalently the injective envelope of ''R''/rad ''R''. The dual of a left module over Λ does not depend on the choice of ''R'' (up to isomorphism). *The transpose Tr(''M'') of a left Λ-module ''M'' is a right Λ-module defined to be the cokernel of the map ''Q'' * → ''P'' *, where ''P'' → ''Q'' → ''M'' → 0 is a minimal projective presentation of ''M''. 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「Artin algebra」の詳細全文を読む スポンサード リンク
|