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Division ring

Discussion in 'Linear and Abstract Algebra' started by xixi, Jun 15, 2010.

  1. Jun 15, 2010 #1
    Let [tex]R[/tex] be a ring . Suppose that [tex]e[/tex] and [tex]f=1-e[/tex] are two idempotent elements of [tex]R[/tex] and we have [tex]R=eRe \oplus fRf[/tex] (direct sum ) and [tex]R[/tex] doesn't have any non-trivial nilpotent element . Set [tex]R_1=eRe[/tex] and [tex]R_2=fRf[/tex] . If [tex]R_1=\{0,e\}[/tex] and [tex]R_2[/tex] is a local ring , then prove that [tex]R_2[/tex] is a division ring . (note that [tex]e[/tex] and [tex]f[/tex] are central idempotents and therefore [tex]fRf=fR[/tex] )
    Last edited: Jun 15, 2010
  2. jcsd
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