Báo cáo nghiên cứu khoa học: "Vành các tự đồng cấu của môđun giả nội xạ và môđun giả xạ ảnh" doc

Σ−
Σ − (1 − C
1
)
R R− A
N N A− X A
ϕ : X −→ N ψ : A −→ N N
N N− R R
R R
R
A− X A
ϕ : N −→ A/X ψ : N −→ A
M
(C
1
) M M
M M
(C
2
) A B M A
M B M
(C
3
) A B M A ∩ B = 0 A ⊕ B
M
(1 − C
1
) M
M
M CS− (1 − C
1

)
M (C
1
) (1 −C
1
) (C
1
) (C
2
) (C
1
) (C
3
)
(C
2
) =⇒ (C
3
)
=⇒ =⇒ =⇒ =⇒ =⇒ (1 − C
1
)
M Σ− Σ−
Σ − (1 − C
1
)) M
(I)
M
(N)
)

(1 − C
1
) I N
M 1− M
M M
1− R 1−
R
R R
R 1−
R R
N A− A− − X
A ϕ : X −→ N ψ : A −→ N N
− N N− N
A− A− − X A
ϕ : N −→ A/X ψ : N −→ A N
N N−
A ⊆ M A ⊆
e
M A ⊆
⊕
M End(M) A
M A M A
M M
Σ−
M 1− S = End(M)
M S 1−
M S 1−
M 1−
M
A M f : A −→ M, (f = 0)

Kerf = 0 f M f
g : M −→ M
Kerf = 0 α = i − f i A
M Kerf ∩ Kerα = 0 M 1− Kerf ∩ Kerα = Kerα
Kerf ∩ Kerα = Kerf Kerf = 0 Kerf ∩ Kerα = Kerα = 0 α
α g : M −→ M gi = α = i − f
f = i − gi = i(id
M
− g) M
ϕ, ψ ∈ S Kerϕ, Kerψ M
Kerϕ ⊆ Kerψ M/Kerψ
∼
=
Imψ ⊆ M M/Kerϕ
∼
=
Imϕ ⊆ M
ϕ
∗
: M/Kerϕ −→ M ϕ
∗
(m + Kerϕ) = ϕ(m) ψ
∗
: M/Kerϕ −→ M
ψ
∗
(m + Kerϕ) = ψ(m) ϕ
∗
M
h ∈ S hϕ

∗
= ψ
∗
m ∈ M hϕ
∗
(m + Kerϕ) = ψ
∗
(m + Kerϕ)
hϕ(m) = ψ(m) hϕ = ψ ψ ∈< ϕ > < ψ >⊆< ϕ >
S
S
S 1−
M 1− M
f : M −→ M/X
f M f f
∗
: M −→
M
α = π − f π
M M/X Imf + Imα = M/X. M 1−
Imf +Imα = Imf Imf +Imα = Imα Imf = M/X Imf +Imα = Imα =
M/X α h hπ = α f = (1 − h)π
M
ϕ, ψ ∈ S Imϕ, Imψ M
Imϕ ⊆ Imψ ψ : M −→ Imψ
ϕ M Imψ M h
ψh = ϕ ϕ ∈< ψ > < ϕ >⊆< ψ > S
S
S 1−
M S = End(M)

M S
M S
M = ⊕
n
i=1
M
i
M
i
1− i = 1, 2, , n End(M) = ⊕
n
i=1
End(M
i
)
Σ
M = ⊕
i∈I
U
i
U
i
M F I A ⊕ (⊕
i∈F
U
i
) ⊆
e
M.
A = M F

A = M A M i ∈ I A ∩ U
i
= 0
F I A ∩ ⊕
i∈F
U
i
= 0 V
1
= ⊕
i∈F
U
i
V
2
= ⊕
i∈K
U
i
K = I\F F A ∩ (V
1
⊕ U
k
) = 0
k ∈ K a ∈ A, a = 0 a = x − u x ∈ V
1
, u ∈ U
k
u = 0
u = a − x ∈ A ⊕ V

1
U
k
∩ (A ⊕ V
1
) = 0 k ∈ K A ⊕ V
1
⊆
e
M
M
M Σ − (1 − C
1
)
M Σ−
(i) ⇒ (ii) (i) M Σ−
M M (C
2
)
M = ⊕
i∈I
U
i
U
i
CS− A
M F ⊆ I
A ⊕ (⊕
i∈F
U

i
) ⊆
e
M.
V
1
= ⊕
i∈F
U
i
V
2
= ⊕
i∈K
U
i
K = I\F p
1
p
2
M V
1
V
2
p
2
|
A
h = p
1

(p
2
|
A
)
−1
p
2
(A) −→ V
1
A = {x +h(x) | x ∈ p
2
(A)} h
V
2
g : B −→ V
1
p
2
(A) ⊆ B ⊆ V
2
h V
2
C = {x + g(x) | x ∈ B} A ⊕ V
1
⊆
e
M p
2
(A) = p

2
(A ⊕ V
1
) ⊆
e
p
2
(M) = V
2
p
2
(M) = V
2
p
2
(A) ⊆
e
B ⊆ V
2
A ⊆
e
C A
M A = C p
2
(A) = B g = h k ∈ K
X
k
= U
k
∩ p

2
(A) X
k
= 0, ∀k ∈ K X
k
A
k
= {x +h(x) | x ∈ X
k
}
X
k
∼
=
A
k
A
k
A A
k
⊆
e
P ⊆ U
k
⊕V
1
A
k
∩V
1

= 0
P ∩ V
1
= 0 p
2
|
P
h
k
= h |
p
2
(A
k
)
h h
k
λ
k
= p
1
(p
2
|
P
)
−1
: p
2
(M) −→ V

1
λ
k
h
k
p
2
(P ) = p
2
(A
k
) p
2
|
P
A
k
⊆
e
P A
k
= P A
k
M V
3
M V
3
= ⊕
i∈L
U

i
L ⊂ I
A
k
⊕ V
3
⊆
e
M.
M (1 − C
1
) A
k
⊆
⊕
M V
3
⊆
⊕
M M
(C
3
) A
k
⊕ V
3
⊆
⊕
M A
k

⊕ V
3
= M V
4
= ⊕
i∈J
U
i
J = I\L
M = A
k
⊕ V
3
= V
4
⊕ V
3
A
k
∼
=
M/V
3
= V
4
⊕ V
3
/V
3
∼

=
V
4
A
k
| J |= 1 A
k
∼
=
U
j
(j ∈ I) X
k
∼
=
A
k
∼
=
U
j
X
k
⊆
⊕
M
M (C
2
) X
k

⊆
e
U
k
⊆
⊕
M X
k
= U
k
, ∀k ∈ F p
2
(A) = V
2
A
∼
=
V
2
= ⊕
i∈K
U
i
A ⊆
⊕
M M CS− M
M Σ− (ii) ⇒ (i) (ii)
M Σ −(1− C
1
) M Σ−

M M
(1 − C
1
) M Σ− M
Σ − (1 − C
1
)
R Σ − (1− C
1
)
R
R
R
R
Σ−
R
(ii) ⇔ (iii) ⇒ (i)
(i) ⇒ (ii) R R
R
= ⊕
n
i=1
R
i
R
i
R
R
R
R

Σ−
Σ−