# 数学代写|抽象代数作业代写abstract algebra代考|Prime Ideals and Maximal Ideals

## 数学代写|抽象代数作业代写abstract algebra代考|Prime Ideals and Maximal Ideals

You’ve probably seen the importance of prime numbers in the study of number theory. Recall that an integer $p$ is prime if its only (positive) divisors are 1 and $p$. An important property of prime numbers is that if $p$ is prime and $p$ divides a product $a b$, then either $p$ divides $a$ or $p$ divides $b$; see Proposition 1.35 . We can rephrase this divisibility property using ideals: if a product $a b$ is in the ideal $p \mathbb{Z}$, then either $a \in p \mathbb{Z}$ or $b \in p \mathbb{Z}$. This version is the right way to generalize the notion of primes to arbitrary rings. ${ }^{11}$
“Integral Domains Are Good; Fields Are Even Better,”
and we ask for which ideals $I$ is the quotient ring $R / I$ an integral domain and for which is it a field.

Definition 3.37. Let $R$ be a commutative ring. An ideal $I$ of $R$ is a prime ideal if $I \neq R$ and if whenever a product of elements $a b \in I$, then either $a \in I$ or $b \in I$.
We observe that if $I$ is a prime ideal, then it also has the following property:
$$a \notin I \text { and } b \notin I \quad \Longrightarrow \quad a b \notin I \text {. }$$
This statement is the contrapositive of, and hence logically equivalent to, the stated definition of prime ideal.

Example 3.38. Let $m \neq 0$ be an integer. The ideal $m \mathbb{Z}$ is a prime ideal if and only if $|m|$ is a prime number in the usual sense.

Example 3.39. Let $F$ be a field. For every $a, b \in F$ with $a \neq 0$, the principal ideal $(a x+b) F[x]$ is a prime ideal. For every $a, b, c \in F$ such that $a \neq 0$ and $b^2-4 a c$ is not equal to the square of an element of $F$, the principal ideal $\left(a x^2+b x+c\right) F[x]$ is a prime ideal. See Exercise 3.52.

The largest possible ideal in a ring $R$ is the entire ring itself. The ideals that are as large as possible without being all of $R$ play an important role.

Definition 3.40. Let $R$ be a commutative ring. An ideal $I$ is called a maximal ideal if $I \neq$ $R$ and if there are no ideals properly contained between $I$ and $R$. In other words, if $J$ is an ideal and $I \subseteq J \subseteq R$, then either $J=I$ or $J=R$.

Example 3.41. Let $p \in \mathbb{Z}$ be a prime number. Then the ideal $p \mathbb{Z}$ is not only a prime ideal, it is also a maximal ideal. This follows by combining Proposition 3.20, which says that $\mathbb{Z} / p \mathbb{Z}$ is a field, with Proposition 3.43 (see below), which says that in general $R / I$ is a field if and only if $I$ is a maximal ideal.

## 数学代写|抽象代数作业代写abstract algebra代考|Vector Spaces and Linear Transformations

The numbers (scalars) that we used for the “arrow-vectors” in Section 4.1 are real numbers. If we instead look at vectors that are pairs $(a, b)$ of numbers and if we add and multiply them by $c$ using the rules (4.1) and (4.2), then we could take $a, b, c$ to be some other sort of number. For example, we could take $a, b, c$ to be complex numbers. More generally, we can use any sort of “numbers” that allow us to add, subtract, multiply, and divide. As we discussed in Section 3.4 , such numbers live in a field; i.e., a commutative ring in which every (non-zero) element has a multiplicative inverse.

Definition 4.1. A field is a commutative ring $F$ with the property that for every $a \in F$ with $a \neq 0$ there is a $b \in F$ satisfying $a b=1$.
Example 4.2. You are already familiar with lots of fields, including $\mathbb{Q}, \mathbb{R}$, and $\mathbb{C}$, known respectively as the fields of rational numbers, real numbers, and complex numbers. For every prime $p$, the ring of integers modulo $p$ is a finite field, denoted $\mathbb{F}_p$ or $\mathbb{Z} / p \mathbb{Z}$; see Proposition 3.20. In Chapter 5 we will see lots of other fields.

In this chapter we fix a field and use it as a basic building block in the definition of a vector space. Subsequently, in Chapter 5 , we will use vector spaces as a fundamental tool to study fields and field extensions.

Definition 4.3. Let $F$ be a field. A vector space with field of scalars $F$, or alternatively an $F$-vector space, is an abelian group $V$ with addition operation + and with a rule for multiplying a vector $v \in V$ by a scalar $c \in F$ to obtain a new vector $c v \in V$. Vector addition and scalar multiplication are required to satisfy the following axioms:
(1) $[$ Identity Law $1 v=v$ for all $v \in V$
(2) [Distributive Law #1] $c\left(\boldsymbol{v}_1+\boldsymbol{v}_2\right)=c \boldsymbol{v}_1+c \boldsymbol{v}_2$ for all $\boldsymbol{v}_1, \boldsymbol{v}_2 \in V$ and all $c \in F$
(3) [Distributive Law #2] $\left(c_1+c_2\right) \boldsymbol{v}=c_1 \boldsymbol{v}+c_2 v$ for all $\boldsymbol{v} \in V$ and all $c_1, c_2 \in F$
(4) [Associative Law]
$$\left(c_1 c_2\right) \boldsymbol{v}=c_1\left(c_2 \boldsymbol{v}\right) \text { for all } \boldsymbol{v} \in V \text { and all } c_1, c_2 \in F$$
The identity element of $V$ is called the zero vector and is denoted by 0 . It should not be confused with $0 \in F$, which is the zero element of the field $F$.

Just as with the axiomatic definitions of groups and rings, there are many basic facts about vector spaces that can be proven directly from the definitions. We list a couple of them here but leave the proofs for you as an exercise.

# 抽象代数代考

## 数学代写|抽象代数作业代写abstract algebra代考|Prime Ideals and Maximal Ideals

(正) 除数是 1 并且 $p$. 素数的一个重要性质是如果 $p$ 是质数和 $p$ 分产品 $a b$ ，那么要么 $p$ 分 裂 $a$ 或者 $p$ 分裂b; 参见提案 1.35。我们可以用理想来重新表述这种可分性: 如果一个产 品 $a b$ 在理想中 $p \mathbb{Z}$, 那么要么 $a \in p \mathbb{Z}$ 或者 $b \in p \mathbb{Z}$. 这个版本是将素数概念推广到任意环的 正确方法。11

“完整的域是好的；领域甚至更好”，

$$a \notin I \text { and } b \notin I \quad \Longrightarrow \quad a b \notin I .$$

## 数学代写|抽象代数作业代写abstract algebra代考|Vector Spaces and Linear Transformations

(1)[身份法 $1 v=v$ 对全部 $v \in V$
(2) [分配律#1] $c\left(\boldsymbol{v}_1+\boldsymbol{v}_2\right)=c \boldsymbol{v}_1+c \boldsymbol{v}_2$ 对全部 $\boldsymbol{v}_1, \boldsymbol{v}_2 \in V$ 和所有 $c \in F$
(3) [分配律 #2] $\left(c_1+c_2\right) \boldsymbol{v}=c_1 \boldsymbol{v}+c_2 v$ 对全部 $\boldsymbol{v} \in V$ 和所有 $c_1, c_2 \in F$
(4) $[$ 结合律 $]$
$\left(c_1 c_2\right) \boldsymbol{v}=c_1\left(c_2 \boldsymbol{v}\right)$ for all $\boldsymbol{v} \in V$ and all $c_1, c_2 \in F$

myassignments-help数学代考价格说明

1、客户需提供物理代考的网址，相关账户，以及课程名称，Textbook等相关资料~客服会根据作业数量和持续时间给您定价~使收费透明，让您清楚的知道您的钱花在什么地方。

2、数学代写一般每篇报价约为600—1000rmb，费用根据持续时间、周作业量、成绩要求有所浮动(持续时间越长约便宜、周作业量越多约贵、成绩要求越高越贵)，报价后价格觉得合适，可以先付一周的款，我们帮你试做，满意后再继续，遇到Fail全额退款。

3、myassignments-help公司所有MATH作业代写服务支持付半款，全款，周付款，周付款一方面方便大家查阅自己的分数，一方面也方便大家资金周转，注意:每周固定周一时先预付下周的定金，不付定金不予继续做。物理代写一次性付清打9.5折。

Math作业代写、数学代写常见问题

myassignments-help擅长领域包含但不是全部: