# Division ring

1. Jun 15, 2010

### xixi

Let $$R$$ be a ring . Suppose that $$e$$ and $$f=1-e$$ are two idempotent elements of $$R$$ and we have $$R=eRe \oplus fRf$$ (direct sum ) and $$R$$ doesn't have any non-trivial nilpotent element . Set $$R_1=eRe$$ and $$R_2=fRf$$ . If $$R_1=\{0,e\}$$ and $$R_2$$ is a local ring , then prove that $$R_2$$ is a division ring . (note that $$e$$ and $$f$$ are central idempotents and therefore $$fRf=fR$$ )

Last edited: Jun 15, 2010