---
# System prepended metadata

title: 'Rings [ch.7]'
tags: [abstract algebra]

---

# Rings [ch.7]
- [Rings [ch3.]](https://hackmd.io/5izknlFIQwCh7GHp8_xLUw)
- Important Note: In this chapter, every ring is **commutative**.
## 7.1. Irreducible Elements and Unique Factorization Domains
- **Definition 7.1: Unit**
![Screenshot 2024-11-15 at 10.05.22 PM](https://hackmd.io/_uploads/ry4LPTVzJg.png)
- **Definition 7.2: Irreducible Elements**
![Screenshot 2024-11-30 at 4.17.15 AM](https://hackmd.io/_uploads/Sy2Hm5vm1l.png)
Note:
    - [irreducible and prime element](https://math.stackexchange.com/questions/4128850/intuition-behind-irreducible-elements-and-prime-elements)
    - In a commutative ring, an element is irreducible if it is neither invertible nor the product of two non-invertible elements, with respect to the multiplication operation on the commutative ring.
    - **When n is the power of 2, then $x^n+1$ is irreducible in rationals.** [[source]](https://math.stackexchange.com/questions/1897285/how-to-prove-x2k1-is-irreducible-over-the-rational-numbers)
- **Definition 7.3: b divides a**
![Screenshot 2024-11-15 at 10.08.22 PM](https://hackmd.io/_uploads/rkXW_pVG1g.png)
- **Definition 7.4: Unique Factorization Domain (UFD)**
![Screenshot 2024-11-30 at 4.17.40 AM](https://hackmd.io/_uploads/SyLDmcw7kl.png)
- **Theorem 7.7: Polynomial Rings and UFD**
![Screenshot 2024-11-15 at 10.11.04 PM](https://hackmd.io/_uploads/rkEKOpNf1l.png)

## 7.2. Euclidean Domains and Principal Ideal Domains
- **Definition 7.8: Principle Ideal Doamin (PID)**
![Screenshot 2024-11-30 at 4.18.07 AM](https://hackmd.io/_uploads/B1xtX5wQ1x.png)
- **Definition 7.9: Euclidean Domain**
![Screenshot 2024-11-15 at 11.19.02 PM](https://hackmd.io/_uploads/B1hvu0EfJg.png)
Note$^5$: 
    - In all of our examples, the size function will have property (2); but we note that if there is a size function σ that satisfies (1), then it is possible to use σ to create a different size function σ′ satisfying both (1) and (2); see Exercise 7.5.
- **Theorem 7.10**
![Screenshot 2024-11-15 at 10.12.21 PM](https://hackmd.io/_uploads/Bk2Aup4Mye.png)
- **Corollary 7.11**
![Screenshot 2024-11-15 at 10.13.00 PM](https://hackmd.io/_uploads/S12eY64Gke.png)
- **Example 7.12**
![Screenshot 2024-11-15 at 11.23.22 PM](https://hackmd.io/_uploads/HJeFYAEzkl.png)
Note:
    - check example 7.13 to see another important example.
- **Proposition: 7.14**
![Screenshot 2024-11-15 at 10.13.40 PM](https://hackmd.io/_uploads/HJrmK6NMJg.png)
- **Proposition 7.15**
![Screenshot 2024-11-15 at 10.14.48 PM](https://hackmd.io/_uploads/B1jDYpVGyg.png)
- **Theorem 7.16**
![Screenshot 2024-11-15 at 10.15.25 PM](https://hackmd.io/_uploads/r1mKFTVfJl.png)
Note:
    - Every Euclidean domain is **PID**.
    
## 7.3. Factorization in Principal Ideal Domains
- **Proposition 7.17**
![Screenshot 2024-11-15 at 10.16.25 PM](https://hackmd.io/_uploads/HJYnta4M1l.png)
- **Corollary 7.18**
![Screenshot 2024-11-15 at 10.16.51 PM](https://hackmd.io/_uploads/BkV15aEzkg.png)
- **Theorem 7.19**
![Screenshot 2024-11-15 at 10.16.58 PM](https://hackmd.io/_uploads/rJ1g9aNf1l.png)
Note:
    - Every PID is a UFD, so Euclidean domain is automatically a unique factorization domain.
- **Corollary 7.20**
![Screenshot 2024-11-15 at 10.17.03 PM](https://hackmd.io/_uploads/BkXzcTEzye.png)

## 7.4. The Chinese Remainder Theorem
- **Theorem 7.22 Chinese Remainder Theorem - Version 1**
![Screenshot 2024-11-15 at 10.18.57 PM](https://hackmd.io/_uploads/BkfqqaVGJe.png)
- **Remark 7.24**
![Screenshot 2024-11-15 at 10.19.29 PM](https://hackmd.io/_uploads/SJf556Nz1l.png)
- **Theorem 7.25 Chinese Remainder Theorem - Version 2**
![Screenshot 2024-11-15 at 10.19.37 PM](https://hackmd.io/_uploads/SkMccaEGJl.png)
- **Lemma 7.26**
![Screenshot 2024-11-15 at 10.19.43 PM](https://hackmd.io/_uploads/S1GcqaEzye.png)
## 7.4.1. An Application of the Chinese Remainder Theorem.
- **Definition 7.27: Euler phi function**
![Screenshot 2024-11-30 at 4.18.53 AM](https://hackmd.io/_uploads/Syfn7cvXye.png)
Example: 
    - $\phi(1)=1, \phi(2)=1, \phi(3)=2, \phi(4)=2, \phi(5)=4, \phi(6)=2, \phi(7)=6.$
- **Corollary 7.29**
![Screenshot 2024-11-15 at 10.21.55 PM](https://hackmd.io/_uploads/r1RbjTNGkl.png)
## 7.5. Field of Fractions
- **Definition 7.30: Field of fractions of R**
![Screenshot 2024-11-30 at 4.19.16 AM](https://hackmd.io/_uploads/Bk46Q9D7Jl.png)
- **Definition 7.32: Field of rational funciton over F**
![Screenshot 2024-11-15 at 10.27.32 PM](https://hackmd.io/_uploads/SyRUhT4f1g.png)
## 7.6. Multivariate and Symmetric Polynomials
- **Definition 7.33: Multivariate Polynomial Ring**
![Screenshot 2024-11-15 at 10.29.22 PM](https://hackmd.io/_uploads/rJN7p64G1e.png)
- **Definition 7.34: The Field if Multivariate Rational Functions**
![Screenshot 2024-11-15 at 10.29.27 PM](https://hackmd.io/_uploads/r14Xp6EG1l.png)
- **Definition 7.35: Permuted Polynomial**
![Screenshot 2024-11-15 at 10.29.35 PM](https://hackmd.io/_uploads/HyEXaTVMkg.png)
- **Definition 7.36: Symmetric Polynomial**
![Screenshot 2024-11-15 at 10.29.43 PM](https://hackmd.io/_uploads/Hk4QaTEfyl.png)
- **Proposition 7.39**
![Screenshot 2024-11-15 at 10.29.48 PM](https://hackmd.io/_uploads/r1E7TpNfJe.png)
- **Definition 7.40: The $k$th elementaary symmetric polynomial**
![Screenshot 2024-11-15 at 10.30.02 PM](https://hackmd.io/_uploads/By4X6pVz1e.png)
Note: [good examples from wikipedia](https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial)
- **Proposition 7.41**
![Screenshot 2024-11-15 at 10.30.11 PM](https://hackmd.io/_uploads/ry4XpaVMkx.png)
![Screenshot 2024-11-15 at 10.30.24 PM](https://hackmd.io/_uploads/SyV7T6VzJx.png)
![Screenshot 2024-11-16 at 12.45.34 AM](https://hackmd.io/_uploads/HJuhnkSM1x.png)
- **Theorem 7.44: Ring of Invariants**
![Screenshot 2024-11-15 at 10.33.05 PM](https://hackmd.io/_uploads/Bk-papNMyx.png)
![Screenshot 2024-11-15 at 10.33.32 PM](https://hackmd.io/_uploads/SJWap6EGJx.png)

## Others
- Every field is a ring.
- $R/I$ is a field if and only if $I$ is a maximal Ideal.
- $I$ is a prime ideal if and only if the quotient ring $R/I$ is an integral domain.
- $I$ is a maximal ideal if and only if the quotient ring $R/I$ is a field.
- Every maximal ideal is a prime ideal, and every prime ideal is a maximal ideal.
- Every Euclidean domain is **PID**.
- Every field is an integral domain.
- Every finite integral domain is a field.
- Every PID is a UFD, so Euclidean domain is automatically a unique factorization domain.
- [Link to Ring [Chapter3]](https://hackmd.io/5izknlFIQwCh7GHp8_xLUw)