Let A be semiprime ring and e a non-zero idempotent.(adsbygoogle = window.adsbygoogle || []).push({});

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.

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# Semiprime ring

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