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## 数学代写|抽象代数代写Abstract Algebra代考|Definition and Examples

In our work with groups, we saw that one way to discover information about a group is to examine its interaction with other groups by way of homomorphisms. It should not be surprising to learn that this concept extends to rings with equally profitable results.
Just as a group homomorphism preserves the group operation, a ring homomorphism preserves the ring operations.

Definition Ring Homomorphism, Ring Isomorphism A ring homomorphism $\phi$ from a ring $R$ to a ring $S$ is a mapping from $R$ to $S$ that preserves the two ring operations; that is, for all $a, b$ in $R$,
$$\phi(a+b)=\phi(a)+\phi(b) \quad \text { and } \quad \phi(a b)=\phi(a) \phi(b) .$$
A ring homomorphism that is both one-to-one and onto is called a ring isomorphism.

As is the case for groups, in the preceding definition the operations on the left of the equal signs are those of $R$, whereas the operations on the right of the equal signs are those of $S$.
Again as with group theory, the roles of isomorphisms and homomorphisms are entirely distinct. An isomorphism is used to show that two rings are algebraically identical; a homomorphism is used to simplify a ring while retaining certain of its features.

A schematic representation of a ring homomorphism is given in Figure 15.1. The dashed arrows indicate the results of performing the ring operations.

## 数学代写|抽象代数代写Abstract Algebra代考|Test for Divisibility by

An integer $\mathrm{n}$ with decimal representation $a_k a_{k-1} \cdots a_0$ is divisible by 9 if and only if $a_k+a_{k-1}+\cdots+a_0$ is divisible by 9 . To verify this, observe that $n=a_k 10^k+a_{k-1} 10^{k-1}+\cdots+a_0$. Then, letting $\alpha$. denote the natural homomorphism from $\mathrm{Z}$ to $Z_9$ [in particular, $\alpha(10)=1$ ], we note that $\mathrm{n}$ is divisible by 9 if and only if
\begin{aligned} 0=\alpha(n) &=\alpha\left(a_k\right)(\alpha(10))^k+\alpha\left(a_{k-1}\right)(\alpha(10))^{k-1}+\cdots+\alpha\left(a_0\right) \ &=\alpha\left(a_k\right)+\alpha\left(a_{k-1}\right)+\cdots+\alpha\left(a_0\right) \ &=\alpha\left(a_k+a_{k-1}+\cdots+a_0\right) \end{aligned}
But $\alpha\left(a_k+a_{k-1}+\cdots+a_0\right)=0$ is equivalent to $a_k+a_{k-1}+\cdots+a_0$ being divisible by 9 .
The next example illustrates the value of the natural homomorphism given in Example $1 .$

## 数学代写|抽象代数代写抽象代数代考|定义和例子

$$\phi(a+b)=\phi(a)+\phi(b) \quad \text { and } \quad \phi(a b)=\phi(a) \phi(b) .$$既为1 – 1又为onto的环同态称为环同构

## 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。