I was interested to know about coalgebraic version of "Idempotents".
So I seached the web and I found the following interesting post :
http://math.stackexchange.com/questions/689322/co-idempotents-algebraic-dual-of-an-idempotent-element.

I this question I try to  look at to coidempotents with a different apparoach.
First we note that an idempotent in  a unital algebra $A$ with multiplication $m:A\otimes A  \to A$, can be reinterpreted  as a linear map $T:A\to A$ which satisfies the following properties:

1. $\;$  $m\circ (T\otimes Id)= T \circ m$

2. $\;$ $T^{2}=T$

The  first condition says that $T$ is  a multiplication operator $T(x)=ex$ for some $e\in A$. The second  condition says that $e$ is  an idempotent. 


So we  naturally reverse the direction of arrows of an algebras and replace the multiplication by comltipication to define  a coidempotent on a coalgebra $(C, \Delta, \epsilon)$ as a linear map $T$ on $A$ which satisfies:

1. $\;$ $(T \otimes Id) \circ  \Delta= \Delta \circ T$ 

2. $\;$ $T^{2}=T$

So  a  coidempotent is  **not**  an element of $C$ but is  an **operator** on $C$ 

A non trivial coidempotent is  an operator $T$ with the above properties which is neither $Id$ nor $0$.

**Questions:**

>1. $\;$ Is there  a nontrivial coidempotent for  the standard comatrix coalgebra or $\mathbb{C}[x]$, with their natural coalgebra structures, respectively ?




>2. $\;$ We know that the structure of the $K$- theory of an algebras $A$ is based on idempotents in $M{\infty}(A)$. Now assume that $C$ is  a  coalgebra. Is  there a natural coalgebra structure on $M_{n}(C)$. If the answer is yes, is there  a natural embedding  which send a coidempotent on $C$ to  a coidempotent in $M_{2}(C)$, and then to higher dimensional matrices(Motivating by the algebraic  process)?


Your answers or comments  are very  appreciated.