|
In mathematics, the associated graded ring of a ring ''R'' with respect to a proper ideal ''I'' is the graded ring: :. Similarly, if ''M'' is a left ''R''-module, then the associated graded module is the graded module over : :. == Basic definitions and properties == For a ring ''R'' and ideal ''I'', multiplication in is defined as follows: First, consider homogeneous elements and and suppose is a representative of ''a'' and is a representative of ''b''. Then define to be the equivalence class of in . Note that this is well-defined modulo . Multiplication of inhomogeneous elements is defined by using the distributive property. A ring or module may be related to its associated graded through the initial form map. Let ''M'' be an ''R''-module and ''I'' an ideal of ''R''. Given , the initial form of ''f'' in , written , is the equivalence class of ''f'' in where ''m'' is the maximum integer such that . If for every ''m'', then set . The initial form map is only a map of sets and generally not a homomorphism. For a submodule , is defined to be the submodule of generated by . This may not be the same as the submodule of generated by the only initial forms of the generators of ''N''. A ring inherits some "good" properties from its associated graded ring. For example, if ''R'' is a noetherian local ring, and is an integral domain, then ''R'' is itself an integral domain. 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「associated graded ring」の詳細全文を読む スポンサード リンク
|