Monday, 5 January 2015

Linguistic Premise Premise of Algebraic Linguistics 3-2 / September 23, 2007

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 aixi, a i = 1
 A,      xi  S
S generates M.
Basis of A module M
Arbitrary x  M
x =  I aiei
M is free A module.

9 <homomorphism>
A module     MN
Map f :  N
f has addition and action A.
x + y ) = f ( x ) + ( ),   ax ) = a f x )   ( a  A, y  )
is homomorphism.
Homomorphism f is bijection.   f is isomorphism.
Set of homomorphism f :  N is expressed by HomA ( M , N )

10 <finitely generate, local ring>
A module     M
M is generated by finite elements { x1, … , xn }       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 Mi
Homomorphism     fi : Mi → Ni
Sequence of module     M1 f1M2f2 fn-2Mn-1fn-1Mn
Ker ( fi+1 ) = Im ( fi )
The sequence is exact aequence.

Tokyo September 23, 2007

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