Feuille d`exercices 2

A I, J A I ⊂J J/I
J A/I (A/I)/(J/I)A/J
A x, y A
A/(x, y)A/(x)y
A/(y)x
Z[i] = {a+bi ∈C;a, b ∈Z}p
(p)Z[i]−1p
Z[i]Z
f:A−→ B
A
B f
f I f A/I 'B
J7→ f(J)K7→ f−1(K)
J A I K B
p⊂A A/p
B A
A I B
m⊂A A/m
B
A f
A I
B

p A p
px = 0 x∈A
Ap
i0< i < p
F:A−→ A, x 7−→ xp
A=Z/pZ
F K
A A
ANil(A)√0AA
Nil(A) = {0}
Nil(A)A
Ar´ed := A/ Nil(A)π:A→Ar´ed
f:A→B B
f0:Ar´ed →B f =f0◦π
A p
A A
Nil(A[X]) = Nil(A)[X]
A x ∈A
x
A u ∈A n ∈A
A u +n A
R n ≥1
Mn(R[X]) →Mn(R)[X]