What is Ideal ?

Anything inside x outside still comes back inside

=> Zero x Anything = Zero

=> Even x Anything = Even

Mathematically,

1. nZ is an Ideal, represented by (n)

Eg. Even subring (2Z) x anything big Ring Z = 2Z = Even

2. (football) Field F is 'sooo BIG' that

(inside = outside)

=> Field has NO Ideal (except trivial 0 and F)

Why was Ideal invented ? because of 'failure" of UNIQUE Primes Factorization" for this case (example):

6 = 2 x 3
but also
$latex 6=(1+\sqrt{-5})(1-\sqrt{-5})$
=> two factorizations !
=> violates the Fundamental Law of Arithmetic which says UNIQUE Prime Factorization

Unique Prime factors exist called Ideal Primes: $latex \mbox{gcd = 2} , \mbox{ 3}$, $latex (1+\sqrt{-5})$, $latex (1-\sqrt{-5}) $

Greatest Common Divisor (gcd or H.C.F.):
For n,m in Z
gcd (a,b)= ma+nb
Example: gcd(6,8) = (-1).6+(1).8=2
(m=-1, n=-1)

Dedekind's Ideals (Ij):
6 =2x3= u.v =I1.I2.I3.I4 ;
$latex u= (1+\sqrt{-5})$
$latex v=(1-\sqrt{-5})$

Let gcd(2,u) = 2M+N.u
M,N in form of $latex a+b\sqrt{-5}$

1. Principal Ideals:
2M = (2) = multiple of 2

2. Ideals (nonPrincipal) = 2M+N.u

3. Ideal prime factors: 6=2 x 3=u.v
Let
I1= gcd(2, u)
I2=gcd(2, v)
I3=gcd(3, u)
I4=gcd(3, v)
Easy to verify (by definition):
I1.I2=(2)
I3.I4=(3)
I1.I3=(u)
I1.I4=(v)
=> Ij are prime & unique factors of 6=I1.I2.I3.I4
=> Fundamental Law of Arithmetic satisfied!

=>Ij "Ideal"-ly exist! hidden behind 'compound' (2,3,u,v) !

Verify : gcd(2, 1+√-5).gcd(2, 1-√-5)=(2) ?

Proof by definition:

[2m+n(1+√-5)][2m'+n'(1-√-5)]
=[2m+n+n√-5 ][2m'+n'-n'√-5]
= 4mm'+2mn'+2nm'+6nn'
= 2(2mm'+mn'+m'n+3nn')
= 2M
= 2 multiples
= (2) = Principal Ideal