Báo cáo nghiên cứu khoa học: "Một số kết quả về V-môđun" ppt

M
M/ Soc M M/ Soc M
M/ Soc M Z(M ) ∩ Z
∗
(M) = 0 M
M M/A A
M Soc M
M M/H H
M H ∩ N N N
M
1.
R
M Rad M Soc M Z(M) E(M)
M M M
E(M) A M A  M
A << M A M A M
M M
1
M M
M
K M m ∈ M − K M
M K m M
K
M
2.
2.1 X M R A B M A 
B 
2.2 N M
N M
N M
M = N. 

2.3 M, X R N M f : N −→ X
L M N ∩ L = ker f N + L = M f
M X 
2.4 M
M/ Soc M
M/ Soc M
M
⇒
⇒ K M m ∈ M − K L
M K m (mR + L)/L
(mR + L)/L M/ Soc M L K L M
L Soc M (mR + L)/L M/L
A/L 0 M/L L
L K m m ∈ A (mR + L)/L ∩ A/L = 0
(mR + L)/L M/L M/L = (mR + L)/L
L M
⇒
X N/ Soc M
M/ Soc M f : N/ Soc M −→ X 0 ker f = K / Soc M X
f 0
(N/ Soc M)/(K/ Soc M)
∼
=
X
N/K
∼
=
X K N N/ Soc M M/ Soc M
N M
K N K M x ∈ N − K L

M K x L
M
M = xR + L = N + L.
N ∩ L = N K = N ∩ L
K/ Soc M = N/ Soc M ∩ L/ Soc M.
f M/ Soc M X X
M/ Soc M
K N K N K
N T M N = K ⊕ T T
T  Soc M  K T = 0 N = K
2.5 M Z
∗
(M) : = {m ∈ M | mR }
M
2.6 M M/ Soc M Z(M) ∩ Z
∗
(M) = 0 M
N
M f 0 N X K = ker f N/K
∼
=
X
K N
K N K M x ∈ N − K L
K x M/ Soc M
M L M
M = xR + L = N + L.
N ∩ L = N N ∩ L K N K = N ∩ L
f M X X M
K N K N

T M N = K ⊕ T N/K
∼
=
T
∼
=
X . T
T T
T + X = E(T ) T E(T ) T ∩ X = Y = 0 T
Y = T T ⊆ X X = E(T) T << E(T )
T X X M
T T T  Z
∗
(M) X
T
∼
=
X T T  Z(M) Z(M)∩Z
∗
(M) = 0
T = 0 N = K
2.7 M A M Soc M
M M/A
X M
M/ Soc M X M/ Soc M A Soc M
X M/A M/A
M = Z Z
A = 6Z ⊃ Soc M = 0 M/A = Z
6
M/ Soc M = Z

2.8 M H M H ∩N
N N M
M
M/H
⇒
⇒ X N M
f : N −→ X K = ker f
(N ∩ H)/(K ∩ H)
∼
=
((N ∩ H) + K)/K  N/K
∼
=
X .
((N ∩ H) + K)/K = N/K
(N ∩ K) + K = N.
(N ∩ H) N K = N
((N ∩ H) + K)/K = 0 N ∩ H = K ∩ H
h : (N + H)/H −→ X h(x + H) = f(x) h
M/H g : M/H −→ X g ◦ i = h
i : (N + H)/H −→ M/H ϕ : M −→ M/H
g ◦ ϕ f X M M
13
313
147
26(3)
20(9)
M
M/ Soc M M/ Soc M
M/ Soc M Z(M) ∩ Z

∗
(M) = 0 M
M M/A A
M Soc M
M M/H H M
H M H ∩ N N
N M