# 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**
- **Definition 7.2: Irreducible Elements**
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**
- **Definition 7.4: Unique Factorization Domain (UFD)**
- **Theorem 7.7: Polynomial Rings and UFD**

## 7.2. Euclidean Domains and Principal Ideal Domains
- **Definition 7.8: Principle Ideal Doamin (PID)**
- **Definition 7.9: Euclidean Domain**
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**
- **Corollary 7.11**
- **Example 7.12**
Note:
- check example 7.13 to see another important example.
- **Proposition: 7.14**
- **Proposition 7.15**
- **Theorem 7.16**
Note:
- Every Euclidean domain is **PID**.

## 7.3. Factorization in Principal Ideal Domains
- **Proposition 7.17**
- **Corollary 7.18**
- **Theorem 7.19**
Note:
- Every PID is a UFD, so Euclidean domain is automatically a unique factorization domain.
- **Corollary 7.20**

## 7.4. The Chinese Remainder Theorem
- **Theorem 7.22 Chinese Remainder Theorem - Version 1**
- **Remark 7.24**
- **Theorem 7.25 Chinese Remainder Theorem - Version 2**
- **Lemma 7.26**
## 7.4.1. An Application of the Chinese Remainder Theorem.
- **Definition 7.27: Euler phi function**
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**
## 7.5. Field of Fractions
- **Definition 7.30: Field of fractions of R**
- **Definition 7.32: Field of rational funciton over F**
## 7.6. Multivariate and Symmetric Polynomials
- **Definition 7.33: Multivariate Polynomial Ring**
- **Definition 7.34: The Field if Multivariate Rational Functions**
- **Definition 7.35: Permuted Polynomial**
- **Definition 7.36: Symmetric Polynomial**
- **Proposition 7.39**
- **Definition 7.40: The $k$th elementaary symmetric polynomial**
Note: [good examples from wikipedia](https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial)
- **Proposition 7.41**
- **Theorem 7.44: Ring of Invariants**

## 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)