Motivated [by this  post](https://mathoverflow.net/questions/259251/a-consecutive-resolution-of-continum-algebras-to-a-simple-continum-algebra) 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?