By Bernard Kolman, David R. Hill
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Additional info for Algebra Lineal (8th Edition). v.Español.
The spaces ϕm (M ) form a chain M = ϕ0 (M ) ⊇ ϕ1 (M ) ⊇ ϕ2 (M ) ⊇ . . with decreasing dimensions. Therefore there exists an index n such that ϕn (M ) = ϕn+1 (M ). Consequently ϕn (M ) = ϕ (M ) for all ≥ n, in particular for = 2n. Denote ψ = ϕn . Then for each m ∈ M there exists an element m ∈ M such that ψ(m) = ψ 2 (m ). Consequently m − ψ(m ) ∈ Ker ψ and hence m = ψ(m ) + (m − ψ(m )) ∈ Im ψ + Ker ψ. Thus M = Im ψ + Ker ψ. If m ∈ Im ψ ∩ Ker ψ then m = ψ(m ) and therefore 0 = ψ(m) = ψ 2 (m ). But ψ induces a bijection ψ(M ) → ψ 2 (M ) = ψ(M ).
If f : M → N is a homomorphism of A-modules, then f (xM ) = xf (M ) ⊆ xN and therefore f induces the K-linear maps fx : xM → xN . The family (fx )x∈S satisfies fy Φ(M )(a) = Φ(N )(a)fx for each a ∈ AS (x, y), that is, (fx )x∈S is a morphism of functors Φ(f ) : Φ(M ) → Φ(N ). Note that A = x,y∈S AS (x, y). Therefore an element a ∈ A can be written in unique way a = x,y∈S ax,y with ax,y ∈ AS (x, y) for all x, y ∈ S. Hence, if F : AS → vec is a K-linear functor, then Ψ(F ) = x∈S F (x) is an Amodule via the multiplication a·(mx )x∈S = a m .
For each pair of objects x, y ∈ C , we choose morphisms (x,y) α1 , . . 2) which become a K-basis in the vector space rad C (x, y)/ rad2 C (x, y) under the canonical projection. Note that by definition C (x, y) is finitedimensional and therefore each space rad C (x, y)/ rad2 C (x, y) is also finitedimensional. A morphism f ∈ C (x, y) is called irreducible if it is radical but does not belong to the radical square. 2) as arrows from x to y. Note that the canonical projection Π : KQC → C is always bijective on the objects.
Algebra Lineal (8th Edition). v.Español. by Bernard Kolman, David R. Hill