Linguistic Premise Premise of Algebraic Linguistics 3-1 / September 23, 2007

Linguistic Premise

 Premise of Algebraic Linguistics 3-1

　　  TANAKA Akio

1 <finitely generated>

Group     G

Subset of G     S

G is generated by finite set S.     G is finitely generated.

2 <ascending chain rule>

Commutative ring     A

Ideal of A   

Ascending chain of I stops finitely.

The situation satisfies ascending chain rule.

3 <descending chain rule>

Commutative ring     A

Ideal of A   

Descending chain of I stops finitely.

The situation satisfies descending chain rule.

4 <maximum element>

Set defined by order     X

Element of X     a, x

x that is a < x does not exist.

a is maximum element.

5 <minimum element>

Set defined by order     X

Element of X     b, x

x that is b > x does not exist.

b is minimum element.

4 <Noetherian ring>

Commutative ring A that satisfies next equivalent conditions is Noetherian ring.

(1) A satisfies ascending chain rule on ideal.

(2) Ideal family of A has maximum element.

(3) Ideal of A is finitely generated.

5 <Artinian  ring>

Commutative ring A that satisfies next equivalent conditions is Artininian ring.

(1) A satisfies descending chain rule on ideal.

(2) Ideal family of A has minimum element.

6 <module>

Additive group     M

Ring     A

M that has action of A is A module.

M satisfies next conditions.

(1) a ( x + y ) = ax + ay,     ( a + b ) x = ax + bx

(2) a ( bx ) = ( ab ) x,     1x = x

7 <direct sum>

A module     M, N

Structure of A module is given by set of product  M ×N. The situation is expressed by M ⊕ N.

Direct sum of n-M, i.e. M ⊕M ⊕….⊕M is expressed by Mⁿ.

Tokyo September 23, 2007

Sekinan Research Field of Language