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Homework Help: Ring theory problem

  1. Jan 26, 2008 #1
    [SOLVED] ring theory problem

    1. The problem statement, all variables and given/known data
    True or false. The product of two nonunits in Z_n may be a unit.


    2. Relevant equations



    3. The attempt at a solution
    If a and b are two non units in Z_n, and ab <= n, then the result is clear, since ab would not be relatively prime to n. But what about if ab > n ?
    Obviously the units form a group under multiplication, but I don't see how that helps.
     
  2. jcsd
  3. Jan 26, 2008 #2
    i don't see how it matters we are in Z_n,

    if a and b are not units in a commutative ring R, then ab is not a unit, if it was then ..some stuff..
     
    Last edited: Jan 26, 2008
  4. Jan 26, 2008 #3
    If ab were a unit, then we would have abc=1, for some c. This implies that a(bc)=1 which implies that a is a unit.

    Is that it?
     
  5. Jan 26, 2008 #4
    of course yea! (just also notice by commutativity (bc)a = 1 too of course, but that's obvious)
     
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