Anneaux factoriels et non factoriels

F
F
A
A
a=up1. . . pru pi
Q[X1, X2, X3, . . . ]
Z
A A\{0}
A∗
a b u ∈A∗a=ub
a∼b
a≥b a |b ,
A\ {0}(A\ {0})/∼
a b A\{0}
A\ {0}(A\ {0})/∼

a b a ≤b a ≥b
A a, b, x A xa xb
a b pgcd(xa, xb) = xpgcd(a, b)
a b xa xb
A
A
A
A a, b A a b
m d ab =md
a b
xy =zt x, y, z, t
A=k[X, Y, Z, T ]/(XY −ZT )
x, y, z, t X, Y, Z, T A
R=k[z, t]⊂A k[Z, T ]A
a=a0+xa1(x) + ya2(y)a0∈R a1∈R[x]a2∈R[y]
A A B =k(X)[Z, T ]
z t A x y
A
xz xy
x z

xa xb e x
xa xb x e d e =xd d
a b e =xd xa xb
d a b d0a b xd0xa xb xd0
pgcd(xa, xb) = e=xd d0d
A
a∈A
a=upα1
1. . . pαr
r=vqβ1
1. . . qβs
sαiβj>0
P={p1, . . . , pr}Q={q1, . . . , qs}a∈A µ(a)
α1+· · · +αr
a=upα1
1. . . pαr
ra µ(a)P∩Q=∅
P Q p1=q1
p1=q1a0
a0=upα1−1
1pα2 2. . . pαr
r=vqβ1−1
1. . . qβs
s
µ(a0)< µ(a)a
a P Q
b=p1q1a b
pgcd(a, b) = pgcd(upα1
1. . . pαr
r, p1q1) = p1pgcd(upα1−1
1. . . pαr
r, q1) = p1
q1pi
pgcd(a, b) = pgcd(vqβ1
1. . . qβs
s, p1q1) = q1pgcd(vqβ1−1
1. . . qβs
s, p1) = q1.
p1=q1a b
m a b ab a
b m d ab =md d a
b
m a b u, v m =ua =vb
ab =uad b =ud ab =vbd a =vd d a b
e a b α, β a =eα b =eβ
eαβ aβ bα a b f
eαβ =mf bα =eαβ =mf =vbf α =vf vd =a=eα =evf
d=ef e d d a b 
X/R
G/H A/I

X G A
R H I A
A
A=k[X]/(P)P
n A
≤n−1x X A
P∈k[X, Y, Z, T ]a∈A
(XY −ZT )P X Y
k[Z, T ]XiYji≥1j≥1
XY ZT
k[Z, T ]
XY
x X
B=k(x)[z, t]f0:k[x, y, z, t]→B x, z, t
y zy/x xy −zt XY −ZT
0f0f:A→B f
f(a)=0 a=a0+xa1(x) + ya2(y)a0+xa1(x) + zt
xa2(zt x)=0
R(x)
x a0xa1(x)
xzt
xa2(zt x)x
a0=a1=a2= 0 a= 0
z=ab A a =a0+xa1(x) + ya2(y)
b=b0+xb1(x)+yb2(y)
B=k(x)[z, t]
z B a
B a ∈k(x)b=a−1z a0+xa1(x) + zt
xa2(zt x)
b0+xb1(x) + zt
xb2(zt x)t
a0∈k , a1∈k[x], a2= 0 b0∈k.z , b1∈k[x].z , b2= 0 .
a x b z x
z=ab a ∈k∗b=a−1z z A
t x y
a0+za1(z) + ta2(t)aik[x, y]
x y z
x y x
A/(y)A/(y)'k[x, y, z, t]/(xy −zt, y)'k[x, z, t]/(zt)
x x z x
A/(z)A/(z)'k[x, y, z, t]/(xy −zt, z)'k[x, y, t]/(xy)
x
xy xy =zt

p q 1p q
xz xy z y pgcd(xz, xy) =
xpgcd(z, y) = x xy =zt pgcd(xz, xy) =
pgcd(xz, zt) = zpgcd(x, t) = z z t
xz xy
x z d = 1
m xz =md =m xy =zt
x z m =xz xy
xz y z x z