Let $R$ be a commutative ring with identity. Recall that an idempotent element $e$ of $R$ is an element a such that $e^2=e$, and a local idempotent is an idempotent a such that $Re$ is a local ring. Also, a von Neumann regular ring is a ring $R$ such that for every $a$ in $R$ there exists an $x$ in $R$ such that $a = axa$.

I am looking for a characterization of local idempotents in a von Neumann regular ring.

Any hint is appreciated

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    $\begingroup$ What sort of characterization do you want? What do you want to do that you can't do just straight from the definitions? $\endgroup$ – Eric Wofsey Aug 31 '19 at 5:18
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    $\begingroup$ Any characterization in term of maximal ideal contaning it or a characterization that is related to von Neumann regularity of $R$. Actually a characterization that localness is hidden in it. $\endgroup$ – A.Rajanda Aug 31 '19 at 5:40

In a von Neumann regular ring, local idempotents are those idempotents $e$ such that $Re$ is a field. In particular, it is also a minimal ideal.


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