Motivated by this post we give the following definition:

Definition: A pair of unital rings $(R,S)$ is called a consecutive pair of rings if they do not have non trivial idempotent and satisfy the following two conditions:

1)There is a surjective morphism $\phi:R\to S$.

2)If a unital ring $W$ without non trivial idempotent possess a surjective sequence of morphisms $R\to W \to S$ then either $W\simeq R$ or $W \simeq S$.

Let $C([0,1])$ be the ring of all complex continuous functions.

Is the pair $(C([0,1]), \mathbb{C})$ a consecutive pair of rings?