$R$ is a ring. Prove that $R/(0_R)\cong R$

Question

$R$ is a ring. Prove that $R/(0_R)\cong R$.

I don't quite understand what $R/(0_R)$ looks like. By definition of quotient ring, it should have cosets $a+(0_R)$ where $a\in R$. So $R/(0_R)$ and $R$ itself really are the same thing?

Answer 1

We can said that this two sets are the same in the following sense: if you remember the definition of the quotient ring, the elements of $R/(0_R)$ are, like you are saying, of the form $$a+(0_R)=\{a\},$$ i.e., are sets consisting only by an element $a\in R$. So the function $\varphi : R\to R/(0_R)$ by $$\varphi(a) =a+ (0_R)=\{a\}$$ is clearly an isomorphism.

Answer 2

Yes, $a \mapsto a+(0_R)$ is the required isomorphism from $R$ to $R/(0_R)$.