Let $M$ be a module over a ring $A$ and $R=Hom_A(M,M)$ its endomorphism ring (with respect to the composition). I need to show these following conditions are equivalent:
- $\alpha = \alpha \beta \alpha$, for some $\beta \in R$
- $\ker(\alpha)$ and im$(\alpha)$ are direct summands of $A$
- the right ideal $\alpha R$ is a direct summand of $R_R$
- the left ideal $R \alpha$ is a direct summand of $_RR$
Searching on the web and on this site, I found out we are talking about Von Neumann regular rings. Also, I found a proof of $1\Leftrightarrow2$. However, I can't prove the other implications. Can someone help me or just give any suitable reference where I can find a proof? Thank you all.