群 (Group)
Def: 一集合 G,與二元運算 * (稱之為乘法),滿足:
    1. 封閉律:對任意 a, b  G ( or [-代表屬於的意思),則 a*b  G
    2. 結合律:對任意 a, b, c  G,則 (a*b)*c = a*(b*c)
    3. 存在單位元素:存在 e  G,使得對任意 g   G,e*g = g*e = g
    4. 存在反元素:對任意 g  G,存在 h   G,使得 g*h = h*g = e
    符合以上四點之 (G,*),稱之為 "群"
    
交換群 (commutative group, abelian group)
Def: 當一群 (G,*),滿足:
    對任意 a, b  G,a*b = b*a 時,
    (G,*) 稱之為交換群

環 (Ring)
Def: 一集合 R,二元運算 + (加法),與二元運算 * (乘法),滿足:
    1. (R,+) 為交換群(加法單位元素記為 0)
    2. 乘法之封閉律:對任意 a, b  R,則 a*b  R
    3. 乘法之結合律:對任意 a, b, c  R,則 (a*b)*c = a*(b*c)
    4. 存在乘法之單位元素:存在 1  R,使得對任意 a  R,a*1 = 1*a = a
    5. 分配律:對任意 a, b, c  R,則 (a+b)*c = a*c+b*c,c*(a+b) = c*a+c*b
    符合以上五點之 (R,+,*),稱之為 "環"

交換環 (commutative ring)
Def: 一環 (R,+,*),滿足:
    對任意 a, b  R,a*b = b*a 時,
    (R,+,*) 稱之為交換環
    
體 (Field)
Def: 一集合 F,二元運算 + (加法),與二元運算 * (乘法),滿足:
    1. (F,+,*) 為交換環
    2. 除了 0 之外的元素均存在乘法反元素:
       對任意 a  G, a != 0,存在 b  G,使得 a*b = b*a = 1
    符合以上二點之 (F,+,*),稱之為 "體"
Stephen H. Friedberg Linear Algebra
    Vector space (or linear space) 有2個operator, said + and *, x and y in V, ! x+y in V
    for element a in F and x in V, ! ax in V
  1. x and y in V, x+y = y+x
  2. x, y and z in V, (x+y)+z = x+(y+z)
  3. There exists a vector denoted 0 such that x+0 = x for each vector x in V
  4. For each vector x in V, there is a vector y in V such that x+y=0
  5. For each vector x in V, 1x=x
  6. For each pair of real numbers a and b and each vector x in V, (ab)x=a(bx)
  7. For each real number a and each pair of vectors x, y in V, a(x+y)=ax+ay
  8. For each pair of real numbers a and b and each vector x in V, (a+b)x=ax+bx

    Define (a1, a2)+(b1, b2) = (a1+b1, a2-b2) and c(a1, a2)=(ca1, 0), 請問是 Vector space嗎?

    Definition: A subset W of a vector space V over a field F is called a subspace of V
    if W is a vector space over F with the operations of addition and scalar multiplication defined on V.

      Subspace
    1. X+Y W whenever xW and yW
    2. axW whenever aF and xW
    3. W has a zero vector
    4. Each vector in W has an additive inverse in W.