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 =2×3= u.v =I1.I2.I3.I4…