---
# System prepended metadata

title: 'Rings [ch.3]'
tags: [abstract algebra]

---

# Rings [ch.3]
- [Ring [ch7.]](https://hackmd.io/vUNEt4KFSvOMm5yZZATseQ)
## 3.1. Introduction to Rings
## 3.2. Abstract Rings and Ring Homomorphisms
- **Definition 3.1: Rings**
![Screenshot 2024-11-15 at 12.15.05 AM](https://hackmd.io/_uploads/HkF-E5Xz1x.png)
Note: 
    - Identity for addition = $0_R$
    - Identity for multiplication = $1_R$
- **Proposition 3.2**
![Screenshot 2024-11-15 at 12.15.35 AM](https://hackmd.io/_uploads/BJgPEqXf1l.png)
- **Definition 3.3: Ring Homomorphism**
![Screenshot 2024-11-15 at 12.22.41 AM](https://hackmd.io/_uploads/ryQAHcXzJl.png)
## 3.3. Interesting Examples of Rings
check the book for ring examples
## 3.4. Some Important Special Types of Rings
- **Definition 3.10: Fields**
![Screenshot 2024-11-15 at 12.25.32 AM](https://hackmd.io/_uploads/ry1F89mMye.png)
Note: a field is a ring
- **Definition 3.13: Integral domain & zero divisor**
![Screenshot 2024-11-15 at 12.28.05 AM](https://hackmd.io/_uploads/By4VDqXGke.png)
- **Proposition 3.15: Cancellation Property for Integral Domains**
![Screenshot 2024-11-15 at 12.29.53 AM](https://hackmd.io/_uploads/H1htD5mfyx.png)

## 3.5. Unit Groups and Product Rings
- **Definition 3.16: Group of Units**
![Screenshot 2024-11-15 at 12.31.10 AM](https://hackmd.io/_uploads/rJWMu97z1x.png)
- **Proposition 3.17**
![Screenshot 2024-11-15 at 12.38.30 AM](https://hackmd.io/_uploads/HkDtF9Qf1l.png)
- **Exampel 3.19**
![Screenshot 2024-11-15 at 12.43.22 AM](https://hackmd.io/_uploads/Bkxnq9mMkg.png)
Note: 
    - Regarding to multiplication, If group of units of $R$ = $R$ with $0_R$ excluded, i.e. $(R^*)=R/\{0\}$), then $R$ is a field.
- **Proposition 3.20**
![Screenshot 2024-11-15 at 12.48.26 AM](https://hackmd.io/_uploads/HkTln9QGyl.png)
- **Definition 3.22: Product of Rings**
![Screenshot 2024-11-15 at 12.49.34 AM](https://hackmd.io/_uploads/By8XhqXz1l.png)
- **Proposition 3.25**
![Screenshot 2024-11-15 at 12.50.39 AM](https://hackmd.io/_uploads/H1Nuh9XzJx.png)
## 3.6. Ideals and Quotient Rings
- **Definition 3.26: Ideals**
![Screenshot 2024-11-15 at 12.52.57 AM](https://hackmd.io/_uploads/HkVl69mfJx.png)
- **Definition 3.27: Principle Ideals**
![Screenshot 2024-11-15 at 12.56.19 AM](https://hackmd.io/_uploads/H15hacmzke.png)
![Screenshot 2024-11-15 at 1.09.46 AM](https://hackmd.io/_uploads/rkCCgoQf1x.png)
- **Definition 3.31: Coset**
![Screenshot 2024-11-15 at 1.10.56 AM](https://hackmd.io/_uploads/HyyVWoQfkg.png)
- **Proposition 3.32**
![Screenshot 2024-11-15 at 1.11.35 AM](https://hackmd.io/_uploads/HysIbj7zke.png)
- **Proposition 3.34**
![Screenshot 2024-11-15 at 1.16.44 AM](https://hackmd.io/_uploads/Hy6uMsXzkg.png)
- **Definition 3.35: Characteristic of a ring**
![Screenshot 2024-11-15 at 1.21.00 AM](https://hackmd.io/_uploads/rkTt7smG1l.png)
Note: 
    - The characteristic of a ring $R$ is the integer $m$ ≥ 0 generating the kernel of the unique homomorphism
    $$\phi: \mathbb{Z}\rightarrow R$$
    $m$ is the generator of the kernel of $\phi$.
- **Theorem 3.36: Frobenius homomorphism of R**
![Screenshot 2024-11-15 at 1.17.13 AM](https://hackmd.io/_uploads/r1e2zsXfkl.png)
## 3.7. Prime Ideals and Maximal Ideals
- **Definition 3.37: Prime Ideal**
![Screenshot 2024-11-15 at 1.22.23 AM](https://hackmd.io/_uploads/BJGyEoXMkg.png)
- **Definition 3.40: Maximal Ideal**
![Screenshot 2024-11-15 at 1.23.10 AM](https://hackmd.io/_uploads/SkH_VjQG1x.png)
Note: 
    - Addition of two distinct ideals of a ring forms the original ring.
    - Every ring has a maximal ideal, see remark 3.45
- **Theorem 3.43**
![Screenshot 2024-11-15 at 1.26.25 AM](https://hackmd.io/_uploads/B1gC4iXzyg.png)
- **Corollary 3.44**
![Screenshot 2024-11-15 at 1.27.09 AM](https://hackmd.io/_uploads/H1FxBjQMkl.png)