You will recall that a ring is a set with two binary operations, addition (denoted by a + b) and multiplication (denoted by ab), provided that it is an ambelian group under addition.
Recall also that an ambelian group is a group that has the commutative property under the given binary operation, in this case -- addition.
In other words, for all a, b, c in the set:
1. a + b = b + a
Moreover, addition is associative (by definition of a group).
2. (a + b) + c = a + (b + c)
Also, addition has identity (by definition of a group).
3. There is an element 0 and are such that a + 0 = a
Then of course, addition has an inverse (by definition of a group).
4. There is an element – a in R such that a + (– a) = 0
Multiplication is also associative.
5. a(bc) = (ab)c
And for all of the rings elements, the left distributive law and the right distributive law hold true.
6. a(b + c) = ab + ac and (b + c)a = ba + ca