Representation of non-primitive idempotent in a finite dimensional algebra

Question

I want to prove the following statement:

For any non-primitive idempotent in a finite dimensional algebra, it can be represented as a sum of orthogonal primitive idempotents.

I tried to decompose the non-primitive orthogonal idepotent term by term, but I cannot show that the subterm is still orthogonal to the others. This procedure may be constructive.

Also, follow the step above, it remains to be shown that the decomposition would stop in finite steps. This claim seems to be relevant to the finiteness of dimension. But I cannot seem to find a way to do this.

Answer 1

I tried to decompose the non-primitive orthogonal idepotent term by term, but I cannot show that the subterm is still orthogonal to the others.

Suppose we split $a+b$ up into $(a_1+\cdots+a_m)+(b_1+\cdots+b_n)$.

We assume $a,b$ are orthogonal, as well as all the pairs $a_i,a_j$ and $b_i,b_j$.

Write $a(b_1+\cdots+b_n)=0$ and right-multiply by $b_i$ to get $ab_i=0$.

Then we may write $(a_1+\cdots+a_m)b_i=0$ and the same trick yields $a_ib_j=0$.

Generalize this trick however you want...

Also, follow the step above, it remains to be shown that the decomposition would stop in finite steps. This claim seems to be relevant to the finiteness of dimension.

Hint: prove orthogonal idempotents must be linearly independent using the above trick.