Linguistic Premise Premise of Algebraic Linguistics 2-1

Linguistic Premise

 Premise of Algebraic Linguistics 2-1

　　  TANAKA Akio

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

Sekinan Research Field of Language