1 <ideal>
Commutative ring A
Subset I ⊂ A
(1) x, y ∈ I ⇒ x-y ∈ I
(2) x ∈ I y ∈ L ⇒ xy = yx ∈ I
I is ideal.
Trivial ideal I = ( 0 ) or I = A
2 <zero divisor>
Commutative ring A
x ∈ A y ∈ A xy = 0
x is zero divisor.
3 <integral domain>
Commutative ring A
A has not zero divisor, except zero element. Zero element is unit of addition.
A is integral domain.
4 <field>
Commutative ring A
A’s element is invertible element , except zero element.
A is field.
Field is integral domain.
4* <proposition on field>
Ring is field. ⇔ A’s ideal is only ( 0 ) or A.
5 <principal ideal>
Ring A
a ∈ A
( a ) = { xa | x ∈ A }
( a ) is principal ideal.
6 <principal ideal domain>
Integral domain A
All the ideals of A are principal domains.
A is principal ideal domain, abbreviated to PID.
7 <Euclidean domain>
Integral domain A
Arbitrary element a ∈ A
N ( a ) ∈ Z
Given conditions
(1) N ( a ) ≥ 0 and N ( a ) = 0 ⇔ a = 0
(2) ∀ a, b ∈ A ( b ≠ 0 ) a = qb + r ( N ( r ) ≺ N ( b ) )
A is Euclidean domain.
7*<Proposition of Euclidean domain>
Ideal of Euclidean domain is principal domain, i.e. Euclid domain is PID.
8 <homomorphism>
Ring A, B
Map φ: A → B
∀a, b ∈ A
φ( a+b ) = φ( a ) + ( b )
φ( ab ) = φ( a )φ( b )
φ( 1 ) = 1
Map φis homomorphism.
9 <isomorphism>
On above 8 <homomorphism>,
Map φis bijection.
Map φis isomorphism.
10 <kernel and image>
Homomorphism of ring φ : A → B
Ker ( φ) = { a ∈ A | φ( a ) = 0 }
Im ( φ) = { φ( a ) | a ∈ A
Ker ( φ) is kernel. Ker ( φ) is A’s ideal.
Im ( φ) is image. Im ( φ) is B’s subring.
11 <quotient ring>
(1)
Ring A’s ideal I
a ∈ A
Quotient class a + I := { a+x | x ∈ I }
(2)
Set of quotient class A/I := { a+I | a ∈ A }
(3)
Set A/I
Given definition
Addition ( a + I ) + ( b + I ) = ( a + b ) + I
Product ( a + I ) ( b + I ) = ab + I
Ring A/I is quotient ring of A by I.
11 <canonical surjection>
Ring A
Ring A’s ideal I
a ∈ A
π( a ) = a + I
i.e.
Homomorphism map π : A → A/I
The map is canonical surjection.
12 <isomorphism theorem>
Ring A
Ring A’s ideal I
Canonical surjection of quotient ring A/I π : A → A/I
Homomorphism φ: A → B
Homomorphism φ= γ o π γ : A/I → B ⇔ Ker φ ⊇ I
12 <prime ideal and maximum ideal>
Ring A
Ideal of ring A I
A/I is integral domain. I is prime ideal.
A/I is field. I is maximum ideal.
Tokyo September 20, 2007
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