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I have a question which is in my ring theory lesson. it's under the topic of distributive lattice and I don't know how to prove it.

Que: If A is a strongly regular ring, then the principle right ideals of A form a boolean algebra wich is isomorphic to the boolean algebra of A.

I want some ideas which clear my mind.

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When $A$ is strongly regular, idempotents are all central, and each principle ideal is generated by a unique idempotent.

One can easily check that the lattice of principal right ideals (they are all ideals, actually) reflects exactly the the partial order $a\leq b\iff ab=a$ on the idempotents of $A$ through the map $e\mapsto eA$.

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  • $\begingroup$ This should be simple to verify. Is there something in particular you need help on? $\endgroup$ – rschwieb Oct 19 '18 at 22:51
  • $\begingroup$ thank you for your response. you're right it was simple but I was confused. $\endgroup$ – Mnik Oct 24 '18 at 7:56

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