can we find an isomorphism between $R$ and $eR$ ,where $R$ is von Neumann regular ring and $e$ is idempotent of $R$? In essay  i am reading, we have an finitely generated ideal, so it is generated by an idempotent element $e\in R$ (since finitely generated ideals generated by an idempotent element in von Neumann regular ring)
Then it is written that ''we can replace $eR$ by $R$''. I can't understand how and why. I am on the opinion that there should be isomorphism between $R$ and $eR$,but couldn't see yet. ($R$ is not domain)
Could anyone help me ?

No, not always.
Look at $R=F_2\times F_2$ where $F_2$ is the field of two elements.
$R$ is von Neumann regular and is clearly not isomorphic to $(1,0)R\cong F_2$.
But on the other hand, if $S=\prod_{i\in \mathbb N} F_2$, there exists a nontrivial idempotent $e$ such that $eS\cong S$ (and there exist other idempotents which do not satisfy that.)
