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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Linear equation over a ring ： ウィキペディア英語版 Linear equation over a ring In algebra, linear equations and systems of linear equations over a field are widely studied. "Over a field" means that the coefficients of the equations and the solutions that one is looking for belong to a given field, commonly the real or the complex numbers. This article is devoted to the same problems where "field" is replaced by "commutative ring", or, typically "Noetherian integral domain". In the case of a single equation, the problem splits in two parts. First, the ideal membership problem, which consists, given a non homogeneous equation :$a\_1x\_1\; +\; \backslash cdots\; +\; a\_kx\_k=b$ with $a\_1,\; \backslash ldots,\; a\_k$ and in a given ring , to decide if it has a solution with $x\_1,\; \backslash ldots,\; x\_k$ in , and, if any, to provide one. This amounts to decide if belongs to the ideal generated by the . The simplest instance of this problem is, for ''k'' = 1 and , to decide if is a unit in . The syzygy problem consists, given ''k'' elements $a\_1,\; \backslash ldots,\; a\_k$ in , to provide a system of generators of the module of the syzygies of $(a\_1,\; \backslash ldots,\; a\_k),$ that is a system of generators of the submodule of those elements $(x\_1,\; \backslash ldots,\; x\_k)$ in ''k'' that are solution of the homogeneous equation :$a\_1x\_1\; +\; \backslash cdots\; +\; a\_kx\_k=0.$ The simplest case, when ''k'' = 1 amounts to find a system of generators of the annihilator of . Given a solution of the ideal membership problem, one obtains all the solutions by adding to it the elements of the module of syzygies. In other words, all the solutions are provided by the solution of these two partial problems. In the case of several equations, the same decomposition into subproblems occurs. The first problem becomes the submodule membership problem. The second one is also called the syzygy problem. A ring such that there are algorithms for the arithmetic operations (addition, subtraction, multiplication) and for the above problems may be called a computable ring, or effective ring. One may also say that linear algebra on the ring is effective. The article considers the main rings for which linear algebra is effective. ==Generalities== To be able of solving the syzygy problem, it is necessary that the module of syzygies is finitely generated, because it is impossible to output an infinite list. Therefore the problems considered here make sense only for Noetherian rings, or at least a coherent ring. In fact, this article is restricted to Noetherian integral domains because of the following result. ''Given a Noetherian integral domain, if there is are algorithms to solve the ideal membership problem and the syzygies problem for a single equation, then one may deduce from them algorithms for the similar problems concerning systems of equations.'' This theorem is useful to prove the existence of algorithms. However, in practice, the algorithms for the systems are designed directly, as it is done for the systems of linear equations over a field. 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「Linear equation over a ring」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.