# Local idempotents in a von Neumann regular rings

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

• What sort of characterization do you want? What do you want to do that you can't do just straight from the definitions? – Eric Wofsey Aug 31 '19 at 5:18
• 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. – A.Rajanda Aug 31 '19 at 5:40

## 1 Answer

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.