** Historical Background**:

**Ideal (理想)** was a by-product by mathematicians in the 350-year proof of the 17CE Fermat’s Last Theorem, wherein they found a violation of the existing “Fundemental Law of Arithmetic” (Unique Prime Factorization) . Since it is a Law, there must be an alternative * ideal* number to satisfy it, hence the birth of the “Ideal”.

Read here: the raison d’être of Ideal : **What is an Ideal ?**

**Note**:

Why Integer (Z) is called “Ring” (Dedekind coined it using the German word “Der Ring”) ? because

{1, 2, … , 11, 12 = 0} is clock number “Z/12Z” like a Ring-shaped Clock 🕜

** Application**:

The ancient “Chinese Remainder Theorem” (aka 韩信点兵 ) since 200 BCE is explained by 19CE Ideal Theory.

[**Solve**] : “The Problem of 6 Professors”

Ideal= “Whateverinsidemultipliesoutside, still comes backinside.”

…

**Ring Examples:**

- Integers Z
- Polynomial with coefficients in Real number , or Complex number, or
**Matrix**(yes!) - Infinite Ring
- Finite Ring (Z/nZ )
- Z/pZ = Field (p is prime)

**Reference**:

33 short videos on the scariest Math subject in universities (France, USA, China, Singapore,… ) “Abstract Algebra” made simple by this charming lecturer.