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