Let A be semiprime ring and e a non-zero idempotent. If Ae is a minimal left ideal then eAe is a division ring. Proof: Suppose that Ae is a minimal left ideal and that exe is different from 0 for x in A. Then $Aexe \subset Ae$ since Ae is an ideal and since Ae is minimal hence Aexe = Ae. Thus there exists a in A such that e = aexe and we get that (eae)(exe)=eae^2xe=eaexe = e^2 =e. The only thing I dont understand is that why is e = aexe? We have that Aexe = Ae so I will say that aexe =ae? but then the rest of proof will not hold? Any suggestions? Thanks.