Monday 5 January 2015

Linguistic Premise Premise of Algebraic Linguistics 2-1 / September 20, 2007

Linguistic Premise

 Premise of Algebraic Linguistics 2-1

    TANAKA Akio

1 <ideal>
Commutative ring     A
Subset     I  A
(1) x      x-y  I
(2) x  I    L      xy yx  I
I is ideal.
Trivial ideal    = ( 0 )  or  A

2 <zero divisor>
Commutative ring      A
x     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 ) = { xa | x  A }
a ) is principal ideal.

6 <principal ideal domain>
Integral domain     A
All the ideals of are principal domains.
A is principal ideal domain, abbreviated to PID.

7 <Euclidean domain>
Integral domain     A
Arbitrary element      A
N ( a )  Z
Given conditions
(1) N ( a )  0 and N ( ) = 0   ⇔  a = 0
(2)  ab  A  ( b  0 )    a = qb +   ( N (  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     AB
Map     φA  B
ab  A
φa+b ) = φa ) + ( b )
φab ) = φ)φb )
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  φ) = 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/:= { a+I | a  A }
(3)
Set A/I
Given definition
Addition   ( a + I ) + ( b + ) = ( a + b ) + I
Product   ( a + I ) ( b + ) = ab + I
Ring A/I is quotient ring of A by I.

11 <canonical surjection>
Ring    A
Ring A’s ideal     I
a  A
πa ) = 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/    π : A  A/I
Homomorphism φA  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

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