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# 数学代写|交换代数代写Commutative Algebra代考|MATH0021 The Noetherian Property

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## 数学代写|交换代数代写Commutative Algebra代考|The Noetherian Property

Definition (6.3.1). – Let A be a ring. One says that an A-module M is noetherian if every non-empty family of submodules of M possesses a maximal element.

Proposition (6.3.2). – Let $\mathrm{A}$ be a ring and let $\mathrm{M}$ be an A-module. The following properties are equivalent:
(i) The module $\mathrm{M}$ is noetherian;
(ii) Every submodule of $\mathrm{M}$ is finitely generated;
(iii) Every increasing family of submodules is stationary (ascending chain condition).

Proof. – (i) $\Rightarrow$ (ii) Let us assume that $\mathrm{M}$ is noetherian, that is, any non-empty family of submodules of $\mathrm{M}$ admits a maximal element and let us show that every submodule of $M$ is finitely generated.

Let $\mathrm{N}$ be a submodule of $\mathrm{M}$ and let us consider the family $\mathscr{S}{\mathrm{N}}$ of all finitely generated submodules of $\mathrm{N}$. This family is non-empty because the null module 0 belongs to $\mathscr{S}{\mathrm{N}}$. By hypothesis, $\mathscr{S}_{\mathrm{N}}$ possesses a maximal element, say, $\mathrm{N}^{\prime}$. By definition, the A-module $\mathrm{N}^{\prime}$ is a finitely generated submodule of $\mathrm{N}$ and no submodule $\mathrm{P}$ of $\mathrm{N}$ such that $\mathrm{N}^{\prime} \subsetneq \mathrm{P}$ is finitely generated.

For every $m \in \mathrm{N}$, the A-module $\mathrm{P}=\mathrm{N}^{\prime}+\mathrm{A} m$ satisfies $\mathrm{N}^{\prime} \subset \mathrm{P} \subset \mathrm{N}$ and is finitely generated; by maximality of $\mathrm{N}^{\prime}$, one has $\mathrm{P}=\mathrm{N}^{\prime}$, hence $m \in \mathrm{N}^{\prime}$. This proves that $\mathrm{N}^{\prime}=\mathrm{N}$, hence $\mathrm{N}$ is finitely generated.

## 数学代写|交换代数代写Commutative Algebra代考|The Artinian Property

Definition (6.4.1). – Let $\mathrm{A}$ be a ring and let $\mathrm{M}$ be a right A-module. One says that $\mathrm{M}$ is artinian if every non-empty family of submodules of $\mathrm{M}$ possesses a minimal element.

Emil Artin (1898-1962) was an Austrian mathematician who worked in algebra, number theory, and topology. He became a professor in Hamburg in 1925. Revoked from this position by the Nazis in 1937, ARтіN and his family took refuge in the United States; he would come back to Germany in 1958.
Dederind had defined for number fields an analogue of the RiemanN zeta function, and ARTIN’s dissertation concerned an analogue of this function for quadratic extensions of the field of rational functions over a finite field $k$, conjecturing a version of the Riemann hypothesis which would be only proved by WeIL in 1948.

In the following years, he and Schrezer introduced “real closed fields”, building an algebraic theory of the real numbers. They characterized them as the only fields whose algebraic closure is a finite extension. He also applied this theory to solve HILBERT’s 17th problem, that a rational function in $\mathbf{R}\left(\mathrm{X}_1, \ldots, \mathrm{X}_n\right)$ which is positive everywhere can be written as a sum of squares of rational functions.

## 数学代写|交换代数代写Commutative Algebra代考|The Noetherian Property

(i) 模块 $M$ 是诺特主义者;
(ii) 每个子模块 $M$ 是有限生成的;
(iii) 每个递增的子模块族都是固定的（升链条件)。

## 数学代写|交换代数代写Commutative Algebra代考|The Artinian Property

Emil Artin (1898-1962) 是奥地利数学家，从事代数、数论和拓扑学研究。他于 1925 年成为汉堡 的教授。1937 年被肭粉撤销这一职位，ARTiN 和他的家人在美国避难；他将在 1958 年回到德国 。Dederind 已经为数字域定义了 RiemanN zeta 函数的模拟，而 ARTIN 的论文涉及该函数的模 拟，用于有限域上有理函数域的二次扩展 $k$, 推测黎曼假设的一个版本，该假设只能在 1948 年由 WelL 证明。

## 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。