Ring Theory
===

- [關於](https://hackmd.io/@0xff07/ByT4ldAS8)

Basics
---

- [Ring Basics](https://hackmd.io/@0xff07/S1Jo0Vv_L)
- [Integral Domain, Division Ring, Field](https://hackmd.io/@0xff07/ByytlMUXd)
- [Subring](https://hackmd.io/@0xff07/ryINZJrX_)
- [Ring Homomorphism](https://hackmd.io/@0xff07/r1vkEIOdL)
- [Example: Quadratic Field](https://hackmd.io/@0xff07/S1uqkNmVd)
- [Example: Quadratic Integer](https://hackmd.io/@0xff07/HJzOLrmEO)

Ideal
---

- [Ideal & Quotient Ring](https://hackmd.io/@0xff07/rysdO8_OU)
- [Ideal Generated by a Set](https://hackmd.io/@0xff07/BkwWnRE7O)
- [Properties of Ideal](https://hackmd.io/@0xff07/Sywy2ohmu)

Isomorphism Theorems
---

- [1st Isomorphism Theorem](https://hackmd.io/@0xff07/H1RFsO2Q_)
- [4th Isomorphism Theorem](https://hackmd.io/@0xff07/ryAvds3Xd)
- [3rd Isomorphism Theorem](https://hackmd.io/@0xff07/rJGORcdVO)
- [2nd Isomorphism Theorem](https://hackmd.io/@0xff07/S1Moxe97O)
- [Chinese Remainder Theorem](https://hackmd.io/@0xff07/BJBIlnnzO)

Rings with Additional Properties
---

- [Maximal Ideal](https://hackmd.io/@0xff07/SJMF852M_)
- [Integral Domain](https://hackmd.io/@0xff07/HJOfgM1Nd)
- [Prime Ideal](https://hackmd.io/@0xff07/SyY61B34d)
- [Irreduciblility and Primes](https://hackmd.io/@0xff07/BJDYKOdQu)
- [Greatest Common Divisors](https://hackmd.io/@0xff07/SkO1RjdQd)

Euclidean Domain
---

- [Euclidean Domain](https://hackmd.io/@0xff07/SkF4pEyX_)
- [Euclidean Algorithm](https://hackmd.io/@0xff07/B1l4L0Hmd)

Principle Ideal Domain
---

- [Principal Ideal Domain](https://hackmd.io/@0xff07/rJ6ypJ8QO)

Unique Factorization Domain
---

- [Unique Factorization Domain](https://hackmd.io/@0xff07/BJ6OWhOQu)
- [Greatest Common Divisor in UFD](https://hackmd.io/@0xff07/H1M7lGy4O)
- [Example: Irreducibles and Primes in Quadratic Integers](https://hackmd.io/@0xff07/HkYXlz1Vu)
- [Example: Primes in Gaussian Integers](https://hackmd.io/@0xff07/B1MkRCCNu)
- [Fermat's Theorem for Sum of Square](https://hackmd.io/@0xff07/HJbYNh-NO)

Polynomial Rings
---

- [Definition of Polynomial Rings](https://hackmd.io/@0xff07/BybOsmJBu)
- [Polynomial Ring on Integral Domains](https://hackmd.io/@0xff07/HJlNJU_Eu)

Polynomial Rings that are UFD
---

- [Field of Fractions](https://hackmd.io/@0xff07/Sk2QevuE_)
- [Gauss Lemma](https://hackmd.io/@0xff07/ry83gxoNd)
- [Polynomial Rings that are UFD](https://hackmd.io/@0xff07/ByNG3BONu)

Irreducibility Criteria
---

- [Irreducibility Criteria](https://hackmd.io/@0xff07/BkMIe5WHu)
- [Eisenstein's Criterion](https://hackmd.io/@0xff07/BkwcsSkm_)

Polynomial Rings over Field
---

- [Polynomial Rings over Field](https://hackmd.io/@0xff07/S1bOWT9Hd)
- [Replacing Coefficients with Isomorphic Field](https://hackmd.io/@0xff07/ByRRFAU8_)