Motivated [by this  post](http://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)For  a  ring $W$ which is  neither isomorphic to $R$ nor to $S$, there is  no  a surjective  sequence  of  morphisms $R\to W \to S$.

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?