Why quotient ring $R/R$ is zero ring $\{ 0\}$? There is already similar question.
Factor rings $R/R$ and $R/0$
Of course, I read it. However I still don't know, how and why the quotient ring $R/R$ is zero ring. So, I ask your help to check my thinking logic.

In my opinion, it seems to be $R/R=R$.
The definition of the quotient ring is $R/P=\{r+P|r \in R\}$.
So, $R/R=\{r+R|r \in R\}$. Does $r+R$ make all of element of $R$?
And if $R$ is a commutative ring with identity, is that something different result?

I think I have a big problem with that logic, but I'm blind to find it. 
I hope your brighter sight.
Thank you in advance.

The elements of the quotient ring $R/P$ are equivalence classes of elements in $R$. That is, two different elements $r_1, r_2 \in R$ produce the same class in $R/P$ if $r_1 - r_2 \in P$. This is equivalent to the definition you wrote above. 
Now $R/0 = R$ because $r_1, r_2 \in R$ are in the same class in $R/0$ only if $r_1 - r_2 = 0$, which means that $r_1 = r_2$. As a result, every element is in its own class, and no two different elements become equal. Thus $R/0$ looks exactly the same as $R$.
However, for any two elements $r_1, r_2 \in R$ we know that $r_1 - r_2 \in R$. Thus in the quotient ring $R/R$, every element is in the same class, so there is only one class! This means we have a ring with only one element -- the zero ring.
