Corollary

Let $A$ be a ring and $a$ a two-sided ideal of $A$. (a) Every left (resp. right, two-sided) ideal of $A/a$ can be written uniquely in the form $b/a$, where $b$ is a left (resp. right, two-sided) ideal of $A$ containing $a$. (b) If $b$ is two-sided, the composite homomorphism $A \to A/a \to (A/a)/(b/a)$ defines when passing to the quotient an isomorphism of $A/b$ onto $(A/a)/(b/a)$.

It suffices to apply Proposition 5 to the canonical morphism of $A$ onto $A/a$.