# 数学代写|数论作业代写number theory代考|Unique Factorization Domains

#### Doug I. Jones

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|数论作业代写number theory代考|Unique Factorization Domains

Let $D$ be a factorization domain. Let $a$ be a nonzero, nonunit element of $D$. Then there exist irreducible elements of $D$ such that
$$a=h_1 h_2 \cdots h_k$$
If $h_1$ and $h_2$ are associates, say $h_2=v h_1$, where $v$ is a unit of $D$, then
$$a=v h_1^2 h_3 \cdots h_k .$$
Repeating this process we eventually obtain a factorization
$$a=w l_1^{k_1} \cdots l_m^{k_m}$$
where $w$ is a unit of $D$, the $k_i$ are positive integers, and the $l_i$ are irreducible elements of $D$ with no two distinct ones being associates. Suppose
$$a=w^{\prime} l_1^{l_1^{\prime}} \cdots l_{m^{\prime}}^{\prime k_{m^{\prime}}^{\prime}}$$
is another such factorization of $a$ into powers of nonassociated irreducible elements of $D$. If $m=m^{\prime}$ and after a possible rearragement of $l_1, \ldots, l_m$, we have
$$l_1 \sim l_1^{\prime}, \ldots, l_m \sim l_m^{\prime} \text { and } k_1=k_1^{\prime}, \ldots, k_m=k_m^{\prime},$$
we say that $a$ has a unique factorization as a product of irreducible elements of $D$.

## 数学代写|数论作业代写number theory代考|Modules

Analogous to the concept of a vector space over a field is that of a module over a ring. All rings are assumed to possess an identity.

Definition 3.4.1 ( $R$-action) Let $R$ be a ring with identity and $M$ an additive Abelian group. A function $\alpha: R \times M \rightarrow M$ is called an $R$-action on $M$ if $\alpha$ has the following properties:
\begin{aligned} & \alpha(r+s, m)=\alpha(r, m)+\alpha(s, m), \ & \alpha(r, m+n)=\alpha(r, m)+\alpha(r, n), \ & \alpha(r, \alpha(s, m))=\alpha(r s, m), \ & \alpha(1, m)=m, \end{aligned}
for all $r, s \in R$ and all $m, n \in M$.
Definition 3.4.2 ( $R$-module) Let $R$ be a ring with identity. An additive Abelian group $M$ together with an $R$-action on $M$ is called an $R$-module.

It would be more accurate to call what we have just defined a left $R$-module. There is a similar definition of a right $R$-module in which the elements of $R$ are written on the right. In this book we will keep to left modules throughout.

If $M$ is an $R$-module with $R$-action $\alpha$ on $M$ we write $\alpha(r, m)$ as $r m$ to keep the notation as simple as possible. With this convention (3.4.1)-(3.4.4) become
\begin{aligned} & (r+s) m=r m+s m, \ & r(m+n)=r m+r n, \ & r(s m)=(r s) m, \ & 1 m=m, \end{aligned}
valid for all $r, s \in R, m, n \in M$. Taking $n=0$ in (3.4.6) and $s=0$ in (3.4.5), we deduce that $r 0=0(r \in R)$ and $0 m=0(m \in M)$. The reader can easily check from the axioms that
$$(-r) m=-(r m)=r(-m)$$
for all $r \in R$ and all $m \in M$.

# 数论作业代写

## 数学代写|数论作业代写number theory代考|Unique Factorization Domains

$$a=h_1 h_2 \cdots h_k$$

$$a=v h_1^2 h_3 \cdots h_k .$$

$$a=w l_1^{k_1} \cdots l_m^{k_m}$$

$$a=w^{\prime} l_1^{l_1^{\prime}} \cdots l_{m^{\prime}}^{\prime k_{m^{\prime}}^{\prime}}$$

$$l_1 \sim l_1^{\prime}, \ldots, l_m \sim l_m^{\prime} \text { and } k_1=k_1^{\prime}, \ldots, k_m=k_m^{\prime},$$

## 数学代写|数论作业代写number theory代考|Modules

\begin{aligned} & \alpha(r+s, m)=\alpha(r, m)+\alpha(s, m), \ & \alpha(r, m+n)=\alpha(r, m)+\alpha(r, n), \ & \alpha(r, \alpha(s, m))=\alpha(r s, m), \ & \alpha(1, m)=m, \end{aligned}

3.4.2 ($R$ -module)设$R$为具有身份的环。加性阿贝尔群$M$与$M$上的$R$ -作用称为$R$ -模块。

\begin{aligned} & (r+s) m=r m+s m, \ & r(m+n)=r m+r n, \ & r(s m)=(r s) m, \ & 1 m=m, \end{aligned}

$$(-r) m=-(r m)=r(-m)$$

## 有限元方法代写

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## 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。

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