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

=> two factorizations !

=> violates the *Fundamental Law of Arithmetic* which says UNIQUE Prime Factorization

Unique Prime factors exist called **Ideal** Primes: , ,

**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 ;

Let gcd(2,u) = 2M+N.u

M,N in form of

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