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. ${ }^{11}$
As additional motivation for this section, we adopt the motto
“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.


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

您可能已经看到素数在数论研究中的重要性。回想一下,一个整数 $p$ 如果它的唯一
(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
我们询问哪些理想 $I$ 是商环 $R / I$ 一个完整的领域,它是一个领域。
定义 3.37。让 $R$ 是交换环。一个理想 $I$ 的 $R$ 是一个素理想如果 $I \neq R$ 如果每当元素的乘 积 $a b \in I$ ,那么要么 $a \in I$ 或者 $b \in I$.
我们观察到如果 $I$ 是素理想,则它还具有以下性质:
a \notin I \text { and } b \notin I \quad \Longrightarrow \quad a b \notin I .
示例 3.38。让 $m \neq 0$ 是一个整数。理想 $m \mathbb{Z}$ 是素理想当且仅当 $|m|$ 是通常意义上的素 数。
示例3.39。让 $F$ 成为一个领域。对于每一个 $a, b \in F$ 和 $a \neq 0$ ,主要理想 $(a x+b) F[x]$ 是素理想。对于每一个 $a, b, c \in F$ 这样 $a \neq 0$ 和 $b^2-4 a c$ 不等于元素的平方 $F$ ,主要理想 $\left(a x^2+b x+c\right) F[x]$ 是素理想。参见练习 3.52。
环中最大可能的理想 $R$ 是整个戒指本身。尽可能大而不是全部的理想 $R$ 扮演一个重要角 色。
定义 3.40。让 $R$ 是交换环。一个理想 $I$ 被称为最大理想,如果 $I \neq R$ 如果两者之间没有 适当的理想 $I$ 和 $R$. 换句话说,如果 $J$ 是一个理想和 $I \subseteq J \subseteq R$ 那么要么 $J=I$ 或者 $J=R$
示例 3.41。让 $p \in \mathbb{Z}$ 是质数。那么理想 $p \mathbb{Z}$ 不仅是素理想,还是最大理想。接下来是结 合提案 3.20,它说 $/ 2 \mathbb{Z}$ 是一个领域,命题 3.43 (见下文),一般来说 $R / I$ 是一个字段 当且仅当 $I$ 是最大理想。

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

我们在第 4.1 节中用于”箭头向量”的数字 (标量) 是实数。如果我们改为查看成对的向 量 $(a, b)$ 数字,如果我们将它们相加并乘以 $c$ 使用规则(4.1)和(4.2),那么我们可以 采取 $a, b, c$ 成为某种其他类型的数字。例如,我们可以采取 $a, b, c$ 成为复数。更一般地 说,我们可以使用任何类型的“数字”来进行加、减、乘、除运算。正如我们在第 3.4 节 中讨论的那样,这些数字存在于一个字段中;即,一个交换环,其中每个 (非零) 元素 都有一个乘法逆元。
定义 4.1。域是一个交换环 $F$ 对于每个 $a \in F$ 和 $a \neq 0$ 有一个 $b \in F$ 令人满意 $a b=1$. 例 4.2。您已经熟悉很多领域,包括 $\mathbb{Q}, \mathbb{R}$ ,和 $\mathbb{C}$ ,分别称为有理数域、实数域和复数 域。对于每个素数 $p$ ,整数环模 $p$ 是一个有限域,表示为 $\mathbb{F} p$ 或者 $\mathbb{Z} / p \mathbb{Z}$; 见提案 3.20。在第 5 章中,我们将看到许多其他字段。
在本章中,我们固定一个字段并将其用作向量空间定义中的基本构建块。随后,在第 5 章中,我们将使用向量空间作为研究域和域扩展的基本工具。
定义 4.3。让 $F$ 成为一个领域。具有标量场的向量空间 $F$ ,或者一个 $F$-向量空间,是阿 贝尔群 $V$ 使用加法运算 + 并使用向量相乘的规则 $v \in V$ 通过一个标量 $c \in F$ 获得一个新 的向量 $c v \in V$. 向量加法和标量乘法需要满足以下公理:
(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 $c_1, c_2 \in F$
的身份元素 $V$ 称为零向量,用 0 表示。它不应该与 $0 \in F$ ,这是字段的零元素 $F$.
正如群和环的公理定义一样,向量空间有许多基本事实可以直接从定义中证明。我们在 这里列出了其中的几个,但将证明留给您作为练习。

数学代写|抽象代数作业代写abstract algebra代考







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