I may be a bit confused but somehow I do not see why the following holds:
Let $R$ be a ring and $I$ an ideal. Then we have the quotient $R/I$. Let $a+I \in R/I$.
Why does it hold that if $a+I = 0 \implies a \in I$? It makes sense to me if think of $\mathbb{Z}/n\mathbb{Z}$. But not in this case, however. How would the formal proof look like? Or is a proof not even necessary and it follows by definition?
Possible explanation
Let's consider $0+I$ which is the set $\{0 + i | \forall i \in I\}=I$, right?
So, I have that $a + I = I$. But this means that $\{a + i | \forall i \in I\} = I$ and since $0 \in I$ we have that $a+0 \in I$ and this is $a \in I$. Is this argumentation correct?