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# 数学代写|交换代数代考Commutative Algebra代写|MA8202 Historic note

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## 数学代写|交换代数代考Commutative Algebra代写|Composition series

Interestingly enough, the discovery of chain conditions for rings and modules, as crystallized in the hands of E. Noether and W. Krull, was an offspring of the early development of group theory. With Lagrange’s and Vandermone’s preliminary incursion, followed by Cauchy’s solid theory of subgroups of the symmetric group, a whole theory of substitutions was going around. Unfortunately, so it appears, Cauchy left the arena quite early, while Galois didn’t live enough to complete his remarkable work. True, no disastrous vacuum took place, with group theory continuing to flash its colors in many other forms, specially in the line of S. Lie’s differential-minded infinite groups-a line of work that had its climax in Klein’s famous Erlangen program. It is said that Klein, being mostly inclined to physics and geometry, benefited from conversation with the algebraist-analyst C. Jordan about the principles of group theory. Thus, one arrives at the crux of the birthplace of the idea of a composition series. It was Jordan that established a fairly complete theory in his famous Traité ([86]). One has to understand the boldness of Jordan’s treatise within an intense period of mathematical output in Europe, where strong-minded scholars like Kronecker, Klein and Dedekind were imposing their influence. Finite group theory didn’t look like a bright prospect to most of them. But there he went, Jordan, writing his second treatise (first one was in analysis). Today’s readers may find the language and notation of the book a bit unsavored, but the style is clear and perfectly readable. One would think that writing on group theory, nearly half a century after Cauchy’s remarkable paper, would bring the style closer to our notation these days. It had to wait a bit more for it to happen, with O. Hölder’s subsequent paper ([77]), about 20 years after Jordan’s treatise came out. Jordan used the French composé for a finite group having proper normal subgroups-hence the subsequent terminology involving the word composition. For example, he called facteurs de composition the orders of the successive quotient groups in a composition series-while nowadays this is the terminology for the quotient groups themselvesand named degré de composition what one calls the length of the group. Of course, the twentieth century metamorphosis to modules had to deal with the fact that these were very rarely finite structures, so the analogy would have to come from the already established theory of vector spaces, where the individual terms have infinite cardinality, but finite dimension. On the bright side, at least in the commutative case, all submodules are trivially normal in the sense of group theory. Therefore, the emphasis on building a nontrivial theory would have to move to imposing finite composition series, thus arriving at the center of Noether’s ideas about chain conditions. For the sake of correctness, it should be remarked that the idea of a composition series for finitely generated modules over a commutative ring essentially aims at having the notion of length, not having the same depth that it has in other fields, such as finite non-Abelian group theory or modules over noncommutative rings and certain categorical generalizations.

## 数学代写|交换代数代考Commutative Algebra代写|Fitting ideals

The remarkable feature of the Fitting ideals is that they give invariants of a finitely generated module $M$ over a Noetherian ring $R$. For each particular structured module, the invariants thus obtained may show under diverse disguise. A systematic use was by Kähler, who employed them to create the so-called Kähler differents (see Section 4.4). These differents are intimately related to others (Dedekind, Noether), only their are easier for computation and relationship with ideal theory. Since $M$ is tantamount to some of its free presentations, understanding the Fitting ideals of $M$ gives a way of putting some of its numerical invariants back in the ring. Although the Noether different was only totally available to the public after her death, thanks to N. Jacobson, it was known to her prior to Fitting’s paper ([58]). The relation between the two differents became the subject of many works in the immediate period thereafter.

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。