Posted on Categories:Commutative Algebra, 交换代数, 数学代写

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

Definition (1.4.1). – Let A be a ring and let I be a subgroup of A (with respect to the addition). One says that $\mathrm{I}$ is a left ideal if for any $a \in \mathrm{I}$ and any $b \in \mathrm{A}$, one has $b a \in \mathrm{I}$. One says that $\mathrm{I}$ is a right ideal if for any $a \in \mathrm{I}$ and any $b \in \mathrm{A}$, one has $a b \in \mathrm{I}$. One says that $\mathrm{I}$ is a two-sided ideal if it is both a left and right ideal.

In a commutative ring, it is equivalent for a subgroup to be a left ideal, a right ideal or a two-sided ideal; one then just says that it is an ideal. Let us observe that in any ring $\mathrm{A}$, the subsets 0 and $\mathrm{A}$ are two-sided ideals. Moreover, an ideal I is equal to A if and only if it contains some unit, if and only if it contains 1

For any $a \in \mathrm{A}$, the set $\mathrm{A} a$ consisting of all elements of $\mathrm{A}$ of the form $x a$, for $x \in \mathrm{A}$, is a left ideal; the set $a \mathrm{~A}$ consisting of all elements of the form $a x$, for $x \in \mathrm{A}$, is a right ideal. When $\mathrm{A}$ is commutative, this ideal is denoted by $(a)$ To show that a subset I of $A$ is a left ideal, it suffices to prove the following properties:
(i) $0 \in \mathrm{I}$;
(ii) for any $a \in \mathrm{I}$ and any $b \in \mathrm{I}, a+b \in \mathrm{I}$;
(iii) for any $a \in \mathrm{I}$ and any $b \in \mathrm{A}, b a \in \mathrm{I}$.
Indeed, since $-1 \in \mathrm{A}$ and $(-1) a=-a$ for any $a \in \mathrm{A}$, these properties imply that $\mathrm{I}$ is a subgroup of $\mathrm{A}$; the third one then shows that it is a left ideal.
The similar characterization of right ideals is left to the reader.

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

Let I and $\mathrm{J}$ be left ideals of $A$; their intersection $\mathrm{I} \cap \mathrm{J}$ is again a left ideal. More generally, the intersection of any family of left ideals of $\mathrm{A}$ is a left ideal of $\mathrm{A}$.

Proof. – Let $\left(\mathrm{I}s\right){s \in \mathrm{S}}$ be a family of ideals of $\mathrm{A}$ and let $\mathrm{I}=\bigcap_{\mathrm{S}} \mathrm{I}s$. (If $\mathrm{S}=\emptyset$, then $\mathrm{I}=\mathrm{A}$.) The intersection of any family of subgroups being a subgroup, $\mathrm{I}$ is a subgroup of $\mathrm{A}$. Let now $x \in \mathrm{I}$ and $a \in \mathrm{A}$ and let us show that $a x \in \mathrm{I}$. For any $s \in S, x \in \mathrm{I}{\mathrm{s}}$, hence $a x \in \mathrm{I}{\mathrm{s}}$ since $\mathrm{I}{\mathrm{s}}$ is a left ideal. It follows that $a x$ belongs to every ideal $\mathrm{I}_s$, hence $a x \in \mathrm{I}$.

We leave it to the reader to state and prove the analogous statements for right and two-sided ideals.

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

(ii) 对于任何 $a \in \mathrm{I}$ 和任何 $b \in \mathrm{I}, a+b \in \mathrm{I}$;
(iii) 对于任何 $a \in \mathrm{I}$ 和任何 $b \in \mathrm{A}, b a \in \mathrm{I}$.

## 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。

Posted on Categories:Commutative Algebra, 交换代数, 数学代写

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

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

Given an $R$-module, one looks for finite sequences of submodules
$$M=M_{0} \supset M_{1} \supset \cdots \supset M_{r}={0} .$$
Note that one can always discard repeated submodules in the sequence, so one assume once for all the condition that $M_{i}$ contains $M_{i+1}$ properly, for every index $i$.

One calls a proper refinement of such a sequence the new sequence obtained by proper insertion of another submodule $M_{i} \supsetneq N \supsetneq M_{i+1}$ between some adjacent terms.
Definition 3.1.3. A sequence as in (3.1.2.1) is called a (finite) composition series if it admits no proper refinements. The number of terms of the series is called its length.
The following easily establishes a class of modules having a composition series.
Proposition 3.1.4. Let $R$ denote an arbitrary commutative ring. Then any $R$-module $M$ which is both Noetherian and Artinian admits a composition series.

Proof. One can assume that $M \neq{0}$. Since $M$ is Noetherian, the family of proper submodules of $M_{0}=M$ admits a maximal element, say, $M_{1} \subset M_{0}$. If $M_{1}={0}$, one is done since $M \supsetneq{0}$ is a composition series. Otherwise, choose a maximal element $M_{2} \mp M_{1}$ in the family of proper submodules of $M_{1}$. Continuing this way, one finds a strictly descending chain of submodules $M=M_{0} \supsetneq M_{1} \supsetneq M_{2} \supsetneq \cdots$ admitting no proper refinements. Since $M$ is Artinian, this chain stabilizes, thus yielding a composition series.

## 数学代写|交换代数代考Commutative Algebra代写|External operations

So far, one has dealt mainly with certain internal behavior of a module. In this regard, just a few of the usual operations with ideals of a ring can be mimicked by submodules $M, N$ of a given module, such as the sum $M+N$ (not a great gain since the outcome is just the smallest submodule containing $M$ and $N$ ).

Other important operations whose results do not leave the ambient module $\mathcal{M}$ requires the intervention of an ideal $I \subset R$, such as the quotient $M: \mathcal{M} I$ or, in the event of a local ring $(R, \mathfrak{m})$, the socle $0:{M} \mathfrak{m}$ of a finitely generated $R$-module $M$. Therefore, it looks pretty urgent to try out some external operations involving two modules. The main ones are the tensor product and the homomorphisms. Thus, given $R$-modules $M$ and $N$, one defines their tensor product $M \otimes{R} N$ and the set of homomorphisms $\operatorname{Hom}_{R}(M, N)$. Both turn out to be $R$-modules, with a big difference: the first is insensitive to the order in which the modules were taken, while the second one gives different results except in the case where $M=N$.

However, nature is often capricious as the module of homomorphisms requires no new definition, while the tensor product does.

One does it by means of a universal property, stressing generators and relations.

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

$$M=M_{0} \supset M_{1} \supset \cdots \supset M_{r}=0 .$$

## 数学代写|交换代数代考Commutative Algebra代写| External operations

$0: M \mathfrak{m}$ 有限生成 $R$-模块 $M$. 因此，尝试一些涉及两个模块的外部橾作看起来非常綮迫。主要的是张量积和同态。因此，给定 $R-$

## 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。