## 数学代写|抽象代数作业代写abstract algebra代考|Unit Groups and Product Rings

This section has two themes. The first is that lurking inside every ring is an interesting group. The second is that we can build bigger rings from smaller rings.

Definition 3.16. Let $R$ be a commutative ring. ${ }^9$ The group of units of $R$ is the subset $R^$ of $R$ defined by $$R^={a \in R \text { : there is some } b \in R \text { satisfying } a b=1} .$$
The group law on $R^$ is ring multiplication. The elements of $R^$ are called units.
Proposition 3.17. The set of units $R^$ is a group, with group law being ring multiplication. Proof. We first check that if $a_1, a_2 \in R^$, then their product $a_1 a_2$ is in $R^$. From the definition of $R^$, we can find $b_1, b_2 \in R^$ satisfying $a_1 b_1=1$ and $a_2 b_2=1$. Then $$\left(a_1 a_2\right)\left(b_1 b_2\right)=\left(a_1 b_1\right)\left(a_2 b_2\right)=1 \cdot 1=1,$$ so $a_1 a_2 \in R^$. It remains to check the three group axioms. First, we note that $1 \in R$ is the identity element; second, the existence of inverses is exactly what defines the elements of $R^$; and third, the associative law for multiplication is one of the ring axioms. Example 3.18. Examples of unit groups include $$\mathbb{Z}^={ \pm 1}, \quad \mathbb{Z}[i]^={ \pm 1, \pm i}, \quad \mathbb{R}[x]^=\mathbb{R}^*$$
Another interesting example is the ring $\mathbb{Z}[\sqrt{2}]={a+b \sqrt{2}: a, b \in \mathbb{Z}}$, whose unit group has infinitely many elements. We leave the proof of these assertions to you; see Exercise 3.28 .
Example 3.19. A ring $R$ is a field if and only if
$$R^={a \in R: a \neq 0}=R \backslash{0},$$ since (3.1) says exactly that every non-zero element of $R$ has a multiplicative inverse. The next example is sufficiently important to merit a formal statement and proof. Proposition 3.20. Let $m \geq 1$ be an integer. Then $$(\mathbb{Z} / m \mathbb{Z})^={a \bmod m: \operatorname{gcd}(a, m)=1} .$$
In particular, if $p$ is a prime number, then $\mathbb{Z} / p \mathbb{Z}$ is a field, often denoted $\mathbb{F}_p$.

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

We recall that in Example 3.4 we constructed the ring $\mathbb{Z} / m \mathbb{Z}$ of integers modulo $m$ by starting with the ring $\mathbb{Z}$ and pretending that two integers $a$ and $b$ are “identical” if their difference $a-b$ is a multiple of $m$. In other words, we defined an equivalence relation on $\mathbb{Z}$ by the rule
$a$ is equivalent to $b$ if $a-b$ is a multiple of $m$,
and we then defined $\mathbb{Z} / m \mathbb{Z}$ to be the set of equivalence classes.
Our goal in this section is to generalize this important construction to arbitrary (commutative) rings. The first step is the generalize the concept of being a “multiple of $m$.”
Definition 3.26. Let $R$ be a commutative ring. An ideal of $R$ is a non-empty subset $I \subseteq R$ with the following two properties:

• If $a \in I$ and $b \in I$, then $a+b \in I$.
• If $a \in I$ and $r \in R$, then $r a \in I$.
One way to create an ideal is to start with a single element of $R$ and take all of its multiples.

Definition 3.27. Let $R$ be a commutative ring, and let $c \in R$. The principal ideal generated by $c$, denoted $c R$ or $(c)$, is the set of all multiples of $c$,
$$c R=(c)={r c: r \in R}$$
We let you verify that $c R$ is an ideal; see Exercise $3.40(\mathrm{a})$.
In some rings, such as $\mathbb{Z}$ and $\mathbb{Z}[i]$ and $\mathbb{R}[x]$, every ideal is a principal ideal, although this is by no means obvious. We will prove these assertions and more in Section 7.2; see also Exercise 3.43 and Theorem 5.21. On the other hand, there are rings such as $\mathbb{Z}[x]$ that have non-principal ideals; see Exercise 3.51 .

# 抽象代数代考

## 数学代写|抽象代数作业代写abstract algebra代考|Unit Groups and Product Rings

$R^{=} a \in R$ : there is some $b \in R$ satisfying $a b=1$.

$$\left(a_1 a_2\right)\left(b_1 b_2\right)=\left(a_1 b_1\right)\left(a_2 b_2\right)=1 \cdot 1=1,$$

$$\mathbb{Z}^{=} \pm 1, \quad \mathbb{Z}[i]= \pm 1, \pm i, \quad \mathbb{R}[x]=\mathbb{R}^*$$

$$R^{=} a \in R: a \neq 0=R \backslash 0$$

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

$a$ 相当于 $b$ 如果 $a-b$ 是的倍数 $m$ ，

• 如果 $a \in I$ 和 $b \in I$ ，然后 $a+b \in I$.
• 如果 $a \in I$ 和 $r \in R$ ，然后 $r a \in I$.
创造理想的一种方法是从一个单一的元素开始 $R$ 并取它的所有倍数。
定义 3.27。让 $R$ 是一个交换环，让 $c \in R$. 产生的主要理想 $c$ ，表示 $c R$ 或者 $(c)$ ，是所有倍 数的集合 $c_r$
$$c R=(c)=r c: r \in R$$
我们让您验证 $c R$ 是一个理想；见练习 $3.40(\mathrm{a})$.
在某些环中，例如 $\mathbb{Z}$ 和 $\mathbb{Z}[i]$ 和 $\mathbb{R}[x]$ ，每个理想都是主要理想，尽管这绝不是显而易见 的。我们将在第 7.2 节中证明这些断言和更多内容；另见练习 3.43 和定理 5.21。另一 方面，还有诸如 $\mathbb{Z}[x]$ 有非主要理想；见习题 3.51。

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