1 close: a? B is another element in the set.
② Law of association: (a? b)？ c = a？ (b？ c)
3 unit yuan: a? E = a and e? a = a
4 inverse? Meta: The inverse of addition is -a, and the inverse of multiplication is 1/a, … (for all elements).
⑤ If it is an integer set, the secondary operation is addition and it is a group (closure is obvious, addition satisfies the associative law, unit element is 0, and inverse element takes inverse number -a).
2. Ring adds a binary operation to Abel group (also called commutative group) (although it is called multiplication, it is different from multiplication of elementary algebra). The algebraic structure is a ring (r,+,...), which needs to satisfy the ring axioms, such as (z,+,? )。 The ring axiom is as follows:
①(R,+) is an Abelian group.
Closure: a+b is another element in the set.
Law of association: (a+b)+c = a+(b+c)
Unit element: the unit element of addition is 0, a+0+a = a, 0+a = a.
Inverse? Meta: The inverse operation of addition is -a, a+ (? a) =(？ A)+a = 0 (for all elements)
Commutation law: A+B = B+A.
② Is (r, ...) a semigroup?
Law of association: (a? b)？ c = a？ (b？ c)
Unit element: the unit element of multiplication is 1, a? 1 = a and 1? a = a
③ Multiplication and addition satisfy the distribution law, and multiplication distribution exceeds addition.
3. In this field, binary operation division is added on the basis of commutative ring, which requires elements (except zero) to be divisible, that is, each non-zero element must have multiplication inverse.
It can be seen that field is an algebraic structure that can be added, subtracted, multiplied and divided (except 0), and it is a generalization of number field and four operations. Integer set, there is no multiplication inverse (1/3 is not an integer), so the integer set is not a domain. Rational number, real number and complex number can form fields, which are called rational number field, real number field and complex number field respectively.