Chain of direct summands and idempotents

Question

Let $M_R$ be a module and $A_1\supseteq A_2\supseteq...$ a chain of direct summands of $M$. I want to find a set of nonzero idempotents $f_i\in End(M)$ with $f_if_j=0$ whenever $i\neq j$.

I know that there are idempotents $f_i\in End(M)$ such that $f_i(M)=A_i$. But I think these are in general not orthgonal. I guess I have to modify my chain, but I don't know how. Any hints?

Answer 1

Observe that (for the $f_i$ you found) $f_if_j=f_jf_i=f_j$ whenever $j\geq i$.

With that in mind, look at $\{f_i-f_{i+1}\mid i\in \mathbb N\}$.