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 ∈ I aiei
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 HomA ( M , N )
10 <finitely generate, local ring>
A module M
M is generated by finite elements { x1, … , xn } 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 Mi
Homomorphism fi : Mi → Ni
Sequence of module M1 →f1M2→f2…→ fn-2Mn-1→fn-1Mn
Ker ( fi+1 ) = Im ( fi )
The sequence is exact aequence.
Tokyo September 23, 2007
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