End. 2. The law of association. 3. What is included? 4. There are opposites.

Then the algebraic structure (g, *) formed by this set and binary operation is called a group.

(2) Abel group: a group whose binary operation also satisfies the commutative law. So Abel group, also called commutative group, is a special group. Binary operations are recorded as "+"

(3) Semigroups: Binary operations defined on sets that satisfy the first two conditions:

1. End. 2. The law of association.

A group must be a semigroup, but a semigroup is not necessarily a group. )

With the above definition, let's take a look at what rings and domains are.

(4) Ring: Let the set R define two binary operations "+"and "*" and satisfy them.

1.(R,+) is an Abelian group.

2.(R, *) is a semigroup.

? 3. The two operations meet the allocation rate, a * (b+c) = a * b+a * c.

Then the algebraic structure formed by the set R and two binary operations is called a ring.

(5) Domain: A semigroup structure in a ring is called a domain if it satisfies the inclusion law and the commutative law. Visible domain is a special kind of ring.

To sum up: the biggest concept is semigroup, group is a subset of semigroup and Abel group is a subset of group. The ring is modified on the basis of Abel group, that is, a binary operation is added to make the set form a semigroup, and the two operations satisfy the distribution rate mentioned above. The last field is a subset of rings, and binary operations that need to be added must also satisfy the inclusion law and the commutative law.