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

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