Structures Algébriques Usuelles

?
?
?
?x−1
? ?
(Z,+)⊂(Q,+)⊂(R,+)⊂(C,+) (Q∗,×)⊂(R∗,×)⊂(C∗,×)
( (P), ◦)⊂( (P), ◦)⊂( (P), ◦)⊂( (P), ◦) (SO(E), ◦)⊂(GO(E), ◦)⊂(GL(E), ◦)
?H⊂H
H6= ; H
H?∀x,y∈H x ?y∈H
H∀x∈H x−1∈H
×
0 0A
×
×1 1A
× ∀ a,b,c∈A a ×(b+c)=a×b+a×c(b+c)×a=b×a+c×a
× ×
(Z,+,×) (M(n), +,×) ( L(E) , +,◦) (R[X], +,×)
×B⊂B
B
B
B×
×I⊂I
I
I× ∀ i∈I∀a∈A a ×i∈I(A×I⊂I)
(K,+,×) 0A6= 1A
×K×
(Q,+,×) (R,+,×) (C,+,×) (R(X), +,×)

K K ×
∀α,β∈K∀x∈Eα.(β.x)=(α×β).x
∀α,β∈K∀x∈E(α+β).x=α.x+β.x
∀α∈K∀x,y∈Eα.(x+y)=α.x+α.y
∀x∈E1K.x=x
(Rn,+, .) (M(n,p), +, .) (F(I,R), +, .) ( L(E,F) , +, .) (RN,+, .) (C[X], +, .)
KF⊂F
F6= ; F30E
F∀α,β∈K∀x,y∈Fα.x+β.y∈F
×
K K ×K
K
×
∀α∈K∀x,y∈Eα.(x×y)=(α.x)×y=x×(α.y)
×K×
(K[X], +,×, .) ( L(E) , +,◦, .) (M(n), +,×, .)
×K K F⊂F
F6= ;
F
F×