11. Anneaux

A a +b ab
A+
a, b, c ∈A
a+b=b+a
(a+b) + c=a+ (b+c)
0∈A a + 0 = a
−a∈A a + (−a) = 0
a, b, c ∈A
a(bc)=(ab)c
a(b+c) = ab +ac (b+c)a=ba +ca
A a, b ∈A ab =ba
1∈A a ∈A
1a=a=a1a
a−1∈A aa−1=a−1a= 1
A
1
a
A U(A)
Z
U(Z) = {1,−1}.
Z2

Znn
U(Zn) = U(n) = {[m] : }.
M2(Z) 2 ×2
Z[x]
C(R) = {f:R→R}
(f+g)(x) := f(x) + g(x) (fg)(x) := f(x)g(x)∀x∈R.
A a, b ∈A
a.0 = 0.a = 0
a(−b)=(−a)b=−(ab)
(−a)(−b) = ab
A
(−1)a=−a
(−1)(−1) = 1
S A
A S
A S ⊂A
A S 6=∅a, b ∈S
ab ∈S
a−b∈S.
{0}A A

{[0],[2],[4]}Z6
n∈NnZ Z
Z[i] = {a+bi :a, b ∈Z}
C
S={f∈C(R) : f(3) = 0}
C(R)
M2×2(Z)
Q(√2) = a+b√2 : a, b ∈Q
R
Q,R C
Q(√2)
Z3[i] = {a+bi :a, b ∈Z3}
i2=−1∈Z3