Let $R$ be a semiperfect ring and $e^2=e\in R$ an idempotent. Prove that $eRe$ and any factor-ring of $R$ is semiperfect too.
I know that $R$ is a semiperfect ring iff $R$ decomposes to a direct sum of right $R$-modules and each of them has got only one maximal submodule. I don't think it's a useful knowledge.
I also know Muller's theorem: it says that $R$ is semiperfect iff $1$ decomposes to a sum of local orthogonal idempotents. This theorem must be the key but I don't understand exactly how to apply it.
Let $1=e_1+\dots+e_n$ be a decomposition in $R$. Then $[1]=[e_1]+\dots+[e_n]$ is a needed decomposition in a factor-ring. But what about $eRe$? Can one show $e=ee_1e+\dots+ee_ne$ is a desired one? How to prove that $\{ee_ie\}_i$ is a family of ortogonal idempotents?
Can you please help me?