why the addition operation of a ring need to be commutative? The definition of a ring requires the addition operation to be commutative. But why it has to be?
 A: Definitions in mathematics don't have to be anything. 
I will repeat this because people say the kinds of things you're saying a lot and so the following kind of thing probably isn't said enough:

Definitions in mathematics can be anything we want them to be.

We could have decided that a ring was a thing with five operations that satisfied thirty axioms if we wanted to. (Actually, we did do something like this: if you unpack the definition of a C*-algebra far enough, it turns out to be a thing with five operations that satisfy twenty-three axioms. But this is beside the point.) Instead, the historically accurate answer is that we chose the ring axioms because they captured a thing that we saw a lot in mathematics and wanted to talk about.
It's a nice exercise to show that commutativity actually follows from the other axioms, in particular it is necessary for distributivity to hold. But this isn't really the point: you could just as well ask why we require multiplication to distribute over addition, and again the historically accurate answer is that we saw this happen a lot and wanted to talk about it. 
If your question is something more like "why is this a good idea?" (which is very different from the question "why does it have to be this way?" because it just doesn't), then one possible answer is the following: in the same way that groups axiomatize symmetries of sets, rings axiomatize symmetries of abelian groups. That is, if $A$ is an abelian group, then the set 
$$\text{End}(A) = \text{Hom}(A, A)$$
of abelian group homomorphisms from $A$ to $A$ naturally forms a ring, with addition given by pointwise addition and multiplication given by composition. This is false if we do not require $A$ to be commutative. Conversely, every ring arises as a subring of the ring of endomorphisms of some abelian group; this is "Cayley's theorem for rings." 
A: You might want to check out the Wikipedia page on near rings, structures that have two operations, addition and multiplication, but the addition is not required to be commutative. Also, the operations are compatible in a limited sense, there is one-sided distribution of the multiplication over the addition.
