Anneaux et corps

1A
A
+×0 1
(A, +,0)
×1
×+∀(a, b, c)∈
A3a×(b+c) = a×b+a×c(b+c)×a=b×a+c×a
(A, +,×,0,1)
ab a.b a ×b A
(A, +,×,0,1)
a∈A
−a a ×
a−1
A a.0 = 0.a = 0 a∈A
a.0 = a.(0+0) = a.0+a.0a.0=0
−a.0 0.a = 0
0 1 0 = 1
A0
a∈A a =a.1 = a.0 = 0 A={0}
A6={0} ⇒ 0A6= 1A
(A, ×,1)
A6={0}
0×
Mn(C) (n, n)
a∈A
×A A∗
A×
(A∗,×,1)
A×
A B
f:A→B
f(A, +) (B, +)
f(1) = 1
(a, a0)∈A2f(aa0) = f(a)f(a0)
f
+A+B×A×B
f(A, ×)
(B, ×)
f f :A∗→B∗
a∈A a0∈A
aa0= 1 f(a)f(a0)=1 f(a)
f:A→B
fker(f) := {a∈A, f(a)=0}f f (A)
A
A0⊂A(A, +) 1
×

Mn(R)Mn(C)
f:A→B
B B 6={0}
f(1) = 1 6= 0
Z Q R
Ck[X]
A
a6= 0 a06= 0
aa0= 0
A
I⊂A
I(A, +)
I A
a∈A i ∈I ai ∈I
a A
{x∈A, ∃y∈A, x =ay},
A A (a)
(a)
A
A A
x1, . . . , xnA
A x1, . . . , xnA
x1, . . . , xn
a1x1+· · · +anxna1, . . . , anA
(x1, . . . , xn)x1, . . . , xn
x1, . . . , xn
x1, . . . , xnA
I A
a I = (a)
f:A→B
B6={0}
A I
A/I
π:A→A/I
f:
A→B I ⊂ker(f)
e
f:A/I →B
f=
e
f◦πker(
e
f) ker(f)A/I
G
H

Z
Z
nZn>0
(Z,+)
nZ
m∈Z
I=nZ Z n>0
Z/nZ
n
n= 0 Z/nZ=Zn= 1
Z/nZ={0}n p
Z/nZab ≡0 (p)
a≡0 (p)b≡0 (p)n>2
p, q
pq ≡0 (n)p q n
K
K
K L
f:K→L K =L
f
K{0}K
I I 6={0}x6= 0
y∈K y =
yx−1x∈I I =K
f:K→L
f K
1 ker(f) = {0}
f:K→L L K
f L K
λ∈K x ∈L λ.x := f(λ)x
K
f:Z→K
f(n) = n.1KN= ker(f)Z
nZn>0
e
f:Z/nZ→KZ/nZ
n p
K
n>0Z→K
0 car(K)
Q,R,C0
pZ/pZ
Fp
p
K K
k⊂K
K
{ki}i∈IK K
G
K k0
K K
K K car(K)=0 k0'Q
car(K) = p > 0k0'Z/pZ