# Ideal in Ring

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 = “Whatever inside multiplies outside, still comes back inside.”

Ring Examples:

1. Integers Z
2. Polynomial with coefficients in Real number , or Complex number, or Matrix (yes!)
3. Infinite Ring
4. Finite Ring (Z/nZ )
5. 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.