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In mathematics, an order in the sense of ring theory is a subring of a ring , such that #''A'' is a ring which is a finite-dimensional algebra over the rational number field # spans ''A'' over , so that , and # is a Z-lattice in ''A''. The last two conditions condition can be stated in less formal terms: Additively, is a free abelian group generated by a basis for ''A'' over . More generally for ''R'' an integral domain contained in a field ''K'' we define to be an ''R''-order in a ''K''-algebra ''A'' if it is a subring of ''A'' which is a full ''R''-lattice.〔Reiner (2003) p.108〕 When ''A'' is not a commutative ring, the idea of order is still important, but the phenomena are different. For example, the Hurwitz quaternions form a maximal order in the quaternions with rational co-ordinates; they are not the quaternions with integer coordinates in the most obvious sense. Maximal orders exist in general, but need not be unique: there is in general no largest order, but a number of maximal orders. An important class of examples is that of integral group rings. Examples:〔Reiner (2003) pp.108–109〕 * If ''A'' is the matrix ring ''M''''n''(''K'') over ''K'' then the matrix ring ''M''''n''(''R'') over ''R'' is an ''R''-order in ''A'' * If ''R'' is an integral domain and ''L'' a finite separable extension of ''K'', then the integral closure ''S'' of ''R'' in ''L'' is an ''R''-order in ''L''. * If ''a'' in ''A'' is an integral element over ''R'' then the polynomial ring ''R''() is an ''R''-order in the algebra ''K''() * If ''A'' is the group ring ''K''() of a finite group ''G'' then ''R''() is an ''R''-order on ''K''() A fundamental property of ''R''-orders is that every element of an ''R''-order is integral over ''R''.〔Reiner (2003) p.110〕 If the integral closure ''S'' of ''R'' in ''A'' is an ''R''-order then this result shows that ''S'' must be the maximal ''R''-order in ''A''. However this is not always the case: indeed ''S'' need not even be a ring, and even if ''S'' is a ring (for example, when ''A'' is commutative) then ''S'' need not be an ''R''-lattice.〔 ==Algebraic number theory== The leading example is the case where ''A'' is a number field ''K'' and is its ring of integers. In algebraic number theory there are examples for any ''K'' other than the rational field of proper subrings of the ring of integers that are also orders. For example in the field extension ''A''=Q(i) of Gaussian rationals over Q, the integral closure of Z is the ring of Gaussian integers Z() and so this is the unique ''maximal'' Z-order: all other orders in ''A'' are contained in it: for example, we can take the subring of the : for which ''b'' is an even number.〔Pohst&Zassenhaus (1989) p.22〕 The maximal order question can be examined at a local field level. This technique is applied in algebraic number theory and modular representation theory. 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「Order (ring theory)」の詳細全文を読む スポンサード リンク
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