Feuille d`exercices 2

A x ∈A
x
C(R,R)
C(R,R)
1−TZ[[T]]
A a, b ∈A1−ab 1−ba
p A p.1A= 0
px = 0 x∈A
Ap
i0< i < p
F:A−→ A, x 7−→ xp
A=Z/pZ
e A e2=e
A A
(Ai)i∈IQi∈IAi
Ai
Z/12ZC(R,R)
e(1−e) (e) = eA
e
A−→ eA ×(1 −e)A
x7→ (ex, (1 −e)x)

k k A k
m:A×A→A k Mn(k)
k=RC(R,R)k
k
a∈A γa:A→A x 7→ ax a
γak γ :A→Endk(A)a7→ γa
kEndk(A)
k A
H=R1⊕Ri⊕Rj⊕RkR
4R R 1
i2=j2=k2=−1 ; ij =−ji =k;ik =−ki =−j;jk =−kj =i.
H R
i j γ :H→EndR(H)'M4(R)
H
H
q=a+bi +cj +dk q =a−bi −cj −dk
qR
q∈Rq=q
R H 1
PHi, j, k
H R
q q =q q2∈R≥0
R H q∈R≥0q≥0
q q =−q
q2∈R≤0
q N(q) = qq N(q)∈RN(q) = N(q)
N(q)=0 q= 0 H
Hx2=−1
H
H C
A
a∈A?n∈(A)a+n∈A?
(A[X]) = (A)[X]

A[X]?=A?+X(A)[X]
A{0}Z/4Z F2[X]/(X2)
A
n n
2 2
Z/4Z F2[X]/(X2)
a3=a
A a3=a a ∈A A
6A={0}
2A3A A 2A+3A=A2A∩3A={0}
2A={0}3A={0}
2A={0}a2=a a ∈A
3A={0}(a+b)3(a−b)3
f:A−→ B
A B f
B A
B A
f f
I A π :A−→ A/I
A I
A/I
A A Rad(A)
A
Rad(A) = {a∈A / ∀b∈A, 1−ab ∈A?}
Rad(A/Rad(A)) = {0}
A
PA I1, . . . , InAP
I1. . . InIk
I A I1I2A
I I ⊂I1, I ⊂I2I1I2⊂I