Posted on Categories:Modern Algebra, 数学代写, 现代代数

# 数学代写|现代代数代考Modern Algebra代写|FIELDS

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## 数学代写|现代代数代考Modern Algebra代写|FIELDS

Because an integral domain has no zero divisors, its set of nonzero elements is closed with respect to multiplication. Therefore, multiplication is an operation on this set of nonzero elements, and it is natural to ask if it yields a group. Because multiplication is required to be associative in a ring, the operation is associative. Because an integral domain has a unity, the operation has an identity element. Thus, with respect to multiplication, the set of nonzero elements of an integral domain can fail to be a group only because of the absence of inverse elements. In the integral domain of integers, for instance, the nonzero elements do not form a group with respect to multiplication because only 1 and -1 have inverses. In the integral domain of rational numbers, however, each nonzero element does have an inverse relative to multiplication. Such integral domains are singled out by the following definition.

Definition. A commutative ring in which the set of nonzero elements forms a group with respect to multiplication is called a field.

Alternatively, a field can be defined as an integral domain in which each nonzero element has an inverse relative to multiplication (Problem 26.11). Another example of a field, besides the ring of rational numbers, is the ring of real numbers. Fields are indispensable to much of mathematics. For example, the field of real numbers is basic in calculus and its applications.
Here are some relationships between classes of rings:
fields $\subset$ integral domains $\subset$ commutative rings $\subset$ rings.
Each class is contained in, but different from, the class that follows it. If we restrict attention to rings having only finitely many elements, however, then the first two classes are the same, because of the following theorem.

## 数学代写|现代代数代考Modern Algebra代写|ISOMORPHISM. CHARACTERISTIC

In Section 18 we met the idea of isomorphism for groups and learned that isomorphic groups are essentially the same-they differ at most in the nature of their elements and operations. A similar idea applies to rings.

Definition. Let $R$ and $S$ be rings. An isomorphism of $R$ onto $S$ is a mapping $\theta: R \rightarrow S$ that is one-to-one and onto and satisfies
$$\theta(a+b)=\theta(a)+\theta(b)$$

and
$$\theta(a b)=\theta(a) \theta(b)$$
for all $a, b \in R$. If there is an isomorphism of $R$ onto $S$, then $R$ and $S$ are said to be isomorphic and we write $R \approx S$.

In the conditions $\theta(a+b)=\theta(a)+\theta(b)$ and $\theta(a b)=\theta(a) \theta(b)$, the operations on the left in each equation are, of course, those of $R$, and the operations on the right are those of $S$. Notice that because of the first of these conditions a ring isomorphism is necessarily an isomorphism of the additive groups of $R$ and $S$. It follows that $\theta(0)=0$ and $\theta(-a)=-\theta(a)$ for each $a \in R$, by translation into additive notation of parts (a) and (b) of Theorem 18.2.
The following example shows that an isomorphism between the additive groups of two rings is not necessarily a ring isomorphism.

# 现代代数代写

## 数学代写|现代代数代考Modern Algebra代写|ISOMORPHISM. CHARACTERISTIC

$$\theta(a+b)=\theta(a)+\theta(b)$$

$$\theta(a b)=\theta(a) \theta(b)$$

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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。