Linguistic Premise Premise of Algebraic Linguistics 3-2

Linguistic Premise

 Premise of Algebraic Linguistics 3-2

　　  TANAKA Akio

8 <free module>

Additive group     M

Subset of M     S

Arbitrary element of M      ∑ⁿ _{i = 1} a_ix_i, a _{i = 1}

∈ A,      x_i ∈ S

S generates M.

Basis of A module M

Arbitrary x ∈ M

x = ∑_i ∈ I a_ie_i

M is free A module.

9 <homomorphism>

A module     M, N

Map f : M → N

f has addition and action A.

f ( x + y ) = f ( x ) + ( y ),   f ( ax ) = a f ( x )   ( ∀a ∈ A, y ∈ M )

f is homomorphism.

Homomorphism f is bijection.   f is isomorphism.

Set of homomorphism f : M → N is expressed by Hom_A ( M , N )

10 <finitely generate, local ring>

A module     M

M is generated by finite elements { x₁, … , x_n }       M is finitely generated.

Ring that has only one maximum ideal is local ring.

11 < Noetherian module, Artinian module>

Applying to Noetherian ring and Artinian module

12 <exact sequence>

A module M_i

Homomorphism     f_i : M_i → N_i

Sequence of module     M₁ →^f1M₂→^f2…→ ^fn-2M_n-1→^fn-1M_n

Ker ( f_i+1 ) = Im ( f_i )

The sequence is exact aequence.

Tokyo September 23, 2007

Sekinan Research Field of Language