Ideal of Ring, Kernel of Group

Last time 1978 in Maths Supérieures (French Classe Préparatoire ) studying Ideal, never understood the “real” meaning except the definition, until I attended the Harvard online course in 2006, which used the MIT Prof Artin’s textbook 《Algebra》, pioneering in the world by using Linear Algebra (Matrices, etc) as the foundation to study Group, Ring, Vector Space etc.

Like any structure in the nature, it has a core (“kernel”) which encapsulates all the essence of the structure : durian kernel, cell kernel, etc.

With kernel we can partition (分类) the whole structure family, eg. 血型={A, B, O, AB} is a “kernel” which can divide all Blood groups into 4.

In a Group structure (only 1 operation, eg. + or *), the Kernel of Group (G) partitions G into “Quotient Group” , denoted as:

G / Ker f

In Ring structure (2 operations : +), the German Hilbert named it “Ideal” (instead of Kernel), also partitions the whole Ring structure. Eg. IDEAL {Even} partitions whole Integer structure family into Even & Odd. The name “Ideal” bcos it is also found ideal number to the uniqueness factorisation satisfying the 《Fundamental Theorem of Arithmetics》

eg. 6 =3*2 = 2*3 (unique factorization ! )

but not true in uniqueness in Complex number, we have also another factorization !

6 = (1+√-5). (1- √-5)

so the Ideal (I) is found as the gcd of these 4 pairs:
I1=gcd ( 2, 1+√-5)
I2 =gcd (2,1- √-5)
I3=gcd (3,1+√-5)
I4=gcd(3,1- √-5)
such that :
6= I1* I2* I3* I4

Ideal is like a “Black-Hole” which sucks everything outside into it to become inside its “core”. Eg. “Even” × anything outside = “Even”, same to “ZERO” Ideal.

A polynomial P(X)is also a Ring Structure (+*, but not / with zero polynomial) has the ideal.
eg.( X^2+1) if it is a factor of P(X), so we can partition P(X) into
P(X) / (X^2+1) sub-Ring structure.

Note : (X^2+1) factor means P(X) has complex root “i”
(= √-1)