Modules et anneaux principaux (TD6)

Z
Z/4Z⊕Z/8Z⊕Z/6Z⊕Z/18Z⊕Z/7Z.
n≥1fZn
fZnf
A A
A A
K n ≥1P∈K[X]n p
i≥1i
P
Mn(K)P
P=X2(X−1)3(X+ 1)
A A
A/M M A
f:M1→M2A
A n AnAn
An
Z
A M, N A ϕ :M→N
Vdϕ:VdM→VdN d ≥1
M=AmN=Ann≥m
ψ:An→An
ψdet ψ A

Z[i]
Z[i]
MZJ M
J2=−1
MZ[i]
MZ
MZ[i]
MZ2rZ
M J 0−Ir
Ir0
x=a+ib Z[i]
Z[i]/(x)a2+b2
pΣ = {a2+b2|a, b ∈N}
Z/pZ Z[i]pΣ
p1 4
A X AXA
X A A(X)
(,) : AX×A(X)→A
(,)A AX∼ −→ HomA(A(X), A)
(,)φ:A(X)→HomA(AX, A)
A=ZX=NE= HomZ(ZN,Z)ψ:E→ZN
ϕ7→ ϕ(δi,j )ji
ϕ(2nan)n(3nbn)n
ϕ∈EZ(N)
(2in)nψZ(N)
φZ
ZNZ
A M A
E1→E2→E3M⊗AE1→M⊗AE2→M⊗AE3
0→E1→E2→E3→0 0 →M⊗AE1→M⊗AE2→
M⊗AE3→0
0→E1→E20→M⊗AE1→M⊗AE2
M A
A M A M
k A1=k[X, Y ]M1(X, Y )A1
A1f:M1×M1→A1/M1f(X, Y )−
f(Y, X)
M1A1
A2=k[T2, T 3]⊆k[T]M2(T2, T 3)A2
M2A2