Một số tính chất của môđun hollow và chiều hollow

K M
M K ∩ L = 0 L
M R M
K M
M
M
M

n
i=1
N
i
N
i
M
udim(M) = n.
K M M K + L = M
L M M M = 0
N M
M
M M
{N
1
, , N
n
} ∩
n
i=1

N
i
M M/N
i
1 ≤ i ≤ n.

Z :
Q :
R :
C :
A ⊆
⊕
B : A B
A →
e
B : A B
A  B : A B
A
∼
=
B : A B
A ⊕ B : A B
ACC (DCC) :
E(M) : M
Soc(M) : M
End(M) : M
u dim(M) : M
Ker(f) Im(f) : f
M
(I)

: ⊕
i∈I
M I M
M
R
(
R
M) : M R
M
n
(S) : n S
Mod R R
Rad(M) : M
J(R) : R
Z(M) : M
R
1 = 0 R
R R M
M = 0 0
M
M
M
M
M M
R M
M
R
Soc(M
R
) S

r
(M)
M
R
Rad(M
R
)
M
R
M
R
M
R
Rad(M
R
) = M Rad(R
R
) = Rad(
R
R) = J(R)
J(R) R
R
R
M
R
Rad(M
R
)  M
R
R M n + 1

M M = M
0
⊃ M
1
⊃ ⊃ M
n
= 0
n M
i−1
/M
i
M
M
M
M lg(M) length(M)
M

→
f
M →
g
M”
M Im(f) = Ker(g)
0 → M

→
f
M →
g
M” → 0

M N R f : M → N
0 → M →
f
N f
M →
f
N → 0 f
0 → M →
f
N → 0 f
f : M → N f

: N → M
f ◦ f

= 1
N
f
f

0 → M

→
f
M →
g
M” → 0
f g
R A ⊂ R
A R r(A) :=

{b ∈ R|ab = 0; ∀a ∈ A} l(A) := {b ∈ R|ba = 0; ∀a ∈ A}
a
A = {a} A
A R
l(A) R
r(A)
A Z(R) R l(A) = r(A)
R
A R l(A) r(A)
R
•
M
1
⊆ M
2
⊆ ⊆ M
n
⊆
n M
n+i
= M
n
i = 1, 2,
•
M
1
⊇ M
2
⊇ ⊇ M
n

⊇
n M
n+i
= M
n
i = 1, 2,
•
• R
R
M R m ∈ M
r
R
(m) →
e
R
R
M M
Z(M
R
)
Z(M
R
) = {m ∈ M|mI = 0 I R }
Z(M
R
) = {m ∈ M|r
R
(m) →
e
R

R
}
Z
r
(R) Z
l
(R) R
M Z(M) = M Z(M) = 0
M M
M
∼
=
A/B B
A
x R
m > 0 x
m
= 0
n x
n
= 0 x A
R n > 0
x
1
, x
2
, , x
n
∈ A x
1

.x
2
x
n
= 0 A
R
A R
n a
1
, a
2
, , a
n
∈
A a
1
.a
2
a
n
= 0 a
n
a
1
= 0
x ∈ R x
2
= x I
g +I R/I
e ∈ R g + I = e + I

I I
x
n
= 0, ∀n ∈ N R/I
e
1
, e
2
R
e
1
.e
2
= e
2
.e
1
= 0 e ∈ R
e = 0
e
1
, e
2
e = e
1
+ e
2
e
1
= 0

e
2
= 0
eR Re
e ∈ R
R
I
r(I) = 0 l(I) = 0
I
1
, I
2
= 0 I
1
.I
2
= 0
x, y ∈ R xRy = 0 x = 0 y = 0
P R R/P
x, y ∈ R xRy ⊆ P x ∈ P y ∈ P
R
R N(R) R
N(R) = 0
N
R
M
R
M
R
N →

e
M N
R
∩ K = 0
K = 0 N
R
M
R
M
R
N  M
K ⊆ M K + N = M K = M
K
A  M ⇔ ∀0 = U ⊂ M A + U ⊂ M
A →
e
M ⇔ ∀0 = U ⊂ M A ∩ U = 0
A  M = 0 ⇒ A = M
A →
e
M = 0 ⇒ A = 0
0  M M →
e
M R M
K M
C M C ∩ K = 0 C
K M K →
e
M
0 K

C →
e
N ⊆ M C = N
C ⊆ N →
e
M N/C →
e
M/C
K M C
K M
K ⊕ C →
e
M
(K ⊕ C)/C →
e
M/C
M
R
= 0
M
R
M
R
M
R
U V M
U ∩ V = 0 M
M
n M
n n

M u dim(M) = n M u dim(M) = n
n
M M
M
M
dim(M) = 0 M = 0
dim(M) = 1 M
N ⊆ M M N
dim(N) ≤ dim(M).
N ⊆ M M dim(N) = dim(M)
N →
e
M
M M

M ⊕M

dim(M ⊕M

) = dim(M)+
dim(M

).
K M
M K  M
L M K + L = M L = M
M N M
M = H ⊕ G H N
(N ∩ G)  M
M

M M
M M = 0 M
M M
N N M
N M
M R M R
M R
N M N
M (N ⊂ Rad(M)) Rad(M)  M
M N  M
M R
M = Rad(M)
M M
M R
M
M
M
(2) (3) N M
M = L⊕K L N (K ∩N)  M
L = M, K = M N = (K ∩ N) ⊂ Rad(M) Rad(M)
M M
X R X
X
∼
=
R/I I
M I
R/I
x
1

, x
2
, , x
n
X X =
Rx
1
+ Rx
2
+ + Rx
n
X
X
Rx
i
i X = Rx
i
X
X
∼
=
R/I I x
i
I M
1
M
2
M
1
+ M

2
= R M
1
M
2
R/I R/I M
1
/I M
2
/I
R/I
I
R/I I
I
M
R/I R I
M M
R/I
M/I R/I
M R M
R
X X
X
∼
=
R/I I
M R
R/I M/I
R/I
M R

M
N M M/N
N  M
M
(1)
(2) (3)
(1) ⇒ (2) : M N
M
N  M M/N
(2) ⇒ (1) : N  M M/N L
M L + N = M
L + N/N  M/N M = L + K K ⊆ M
M/N = L+K/N = L+N/N +K+N/N L+N/N  M/N
M = K + N N  M M = K M
(1) ⇒ (3) : M M/N
M M/N
M/N
(3) ⇒ (1) : L M
M = L +K K ⊆ M M/L ∩K
∼
=
M/L⊕M/K
(3) M/L ∩ K M/L = 0
M/K = 0 L
M M/L = 0 M/K = 0 M = K
M
M R M
K N
f : K → M g : M → N g ◦ f f g
M

g ◦ f
g M = Im(f) + Ker(g)
M Ker(g)  M M = Im(f)
f
K L
M M = K + L
0 −→ K −→
i
M −→
π
M/L −→ 0 i
π π ◦ i
i K = M M
M R M
M
M
R M
N M
M m ∈ M m /∈ N
Rm = M. Rm
M N M Rm ⊆ N
M
M L
M L + K = M K
M K = nM K L M
N L K
N M = K + L ⊆ N N = M
K = M M
M
M N L

M L N
L + N = M M N = M
N
M N M
M
M
{N
λ
}
Λ
M
λ ∈ Λ
F ⊆ Λ \ {λ} N
λ
+ ∩
i∈F
N
i
= M
M
M
M
M
N K M N
K M M = N +K N ∩K  M
M
M M
M n
f : M → ⊕
n

i=1
M
i
Ker(f)  M
M
i
⊆ M f
hdim(M) = n M = 0 hdim(M) = 0 M
hdim(M) = ∞
M M

M ⊕ M

dim(M ⊕ M

) = dim(M) + dim(M

).
H
H = H
1
⊕ H
2
⊕ ⊕ H
n
H
i
, i = 1, 2, , n
H hdim(H) = hdim(H
1

) + hdim(H
2
) + + hdim(H
n
).
H
i
H hdim(H) ≥
hdim(H
i
) N K H
{P
1
/N, P
2
/N, , P
k
/N} H/N
P
1
, P
2
, , P
k
H
hdim(H/N) ≤ hdim(H) hdim(H
i
) = ∞
H
i

hdim(H) = ∞
i ∈ {1, 2, , k}
hdim(H
i
) = n
i
< n. 1 ≤ i ≤ k
{H
i
1
, H
i
2
, , H
i
n
i
} H
i
∩
n
i
j=1
H
j
 H
i
H
i
/H

i
j
j = 1, 2, , n
i
i
0
∈ {1, 2, , k} j
0
∈ {1, 2, , n
i
0
} H
1i
0
j
0
H = ⊕
k
i=1
H
i
⊕
k
i=1
A
i
 A
i
0
= H

i
0
j
0
A
j
= H
j
j = i
0
{H
1i
0
j
0
: 1 ≤ i
0
≤ k, 1 ≤ j
0
≤ n
i
0
}
Σ
k
i=1
n
i
H
∩

k
i=1
H
i
 H H/H
1i
0
j
0
∼
=
H
i
0
/H
i
0
j
0
i
0
, j
0
. hdim(H) = Σ
k
i=1
n
i
.
M R

N M hdim(M) = hdim(M/N)+hdim(N)
M
hdim(M) = hdim(M/N) + hdim(N)
N K M N K
M M
M
N ⊆ M M
1
+ (M
2
+
N) (M
2
+ N)
M N M
M N
K K + N = M (K ∩ N)  M
hdim(M) = hdim(M/(K ∩N)) = hdim(K/(K ∩N)⊕(N/(K ∩N))) =
hdim(K/(K ∩ N) +hdim(N/(K ∩N))) = hdim(M/N) +hdim(M/K)
hdim(M) = hdim(M/N) + hdim(M/K)
hdim(N) = hdim(M/K) = hdim(N/(N ∩ K))
hdim(N) N ∩ K  N N
K M M
hdim(M) =
hdim(K) + hdim(L) K L M
M R
K L M M = K + L K L
hdim(M) = hdim(K) + hdim(L)
K L
M K ∩ L  K hdim(K) = hdim(K/(K ∩ L)) M

N M
N  M hdim(M) = hdim(M/N)
L M M
N ⊆ M N M hdim(M) =
hdim(M/N) + hdim(N) hdim(M) = hdim(M/L) + hdim(L)
hdim(M) = hdim(K) + hdim(L).
•
•