# 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**
Note:
- Identity for addition = $0_R$
- Identity for multiplication = $1_R$
- **Proposition 3.2**
- **Definition 3.3: Ring Homomorphism**
## 3.3. Interesting Examples of Rings
check the book for ring examples
## 3.4. Some Important Special Types of Rings
- **Definition 3.10: Fields**
Note: a field is a ring
- **Definition 3.13: Integral domain & zero divisor**
- **Proposition 3.15: Cancellation Property for Integral Domains**

## 3.5. Unit Groups and Product Rings
- **Definition 3.16: Group of Units**
- **Proposition 3.17**
- **Exampel 3.19**
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**
- **Definition 3.22: Product of Rings**
- **Proposition 3.25**
## 3.6. Ideals and Quotient Rings
- **Definition 3.26: Ideals**
- **Definition 3.27: Principle Ideals**
- **Definition 3.31: Coset**
- **Proposition 3.32**
- **Proposition 3.34**
- **Definition 3.35: Characteristic of a ring**
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**
## 3.7. Prime Ideals and Maximal Ideals
- **Definition 3.37: Prime Ideal**
- **Definition 3.40: Maximal Ideal**
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**
- **Corollary 3.44**