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.

What makes a good Conjecture?

If you can’t prove, then makes an outrageous conjecture!

Examples:

  1. Fermat’s Last Theorem (Conjecture for 380 years until proven as “Theorem” in 1994)
  2. Poincare “Conjecture” = (Theorem proven by Grigoli Perelman )
  3. Goldbach’s Conjecture (best proof by 陈景润 “1+2”)
  4. Yang-Mills Conjecture (7 Millennia Clay-Prize Problems)
  5. Collatz Conjecture

https://www.quantamagazine.org/the-subtle-art-of-the-mathematical-conjecture-20190507/