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Irreducible central elements generate maximal ideals?

  1. Nov 24, 2013 #1
    http://img17.imageshack.us/img17/3937/vjj6.png [Broken]

    I don't understand the very last bit, where it says a and b are central and so c is, hence a is not irreducible. Basically what replaces the "hence"? I can't seem to figure out the link between c being central meaning a isn't irreducible.

    a being irreducible means it isn't a unit, hence D[x]a is proper but if it is not maximal it sits inside an ideal D[x]b where b is a central non-unit.

    Hence a = cb for some c. Since a and b are non-units, to be irreducible c must be a unit. But then c being central contradicts it being a unit? Why? There are central units contained in H[x] for instance (the ring of quaternions), a real number is a central unit.
     
    Last edited by a moderator: May 6, 2017
  2. jcsd
  3. Nov 24, 2013 #2
    Hmm, left it for a few hours and came back to it. Noticed what I was doing wrong, not noticing that it was talking about irreducibility over F[x], instead of the whole ring D[x]. Therefore c cannot be a unit otherwise D[x]a does not sit properly in D[x]b, but since c is central and b is central, a = cb is a product of two non-unit elements of F[x]. Hence a is not irreducible over F[x].
     
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