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Factor Rings and Ideals?

  1. Mar 22, 2009 #1
    Hi so this isnt homework its in my book , i just dont get it they skipped this step


    Let R=Z(i) be the ring of gaussian integers and let A=(2+i)R denote the ideal of all multiples of 2+i Describe the cosets of R/A

    im just having trouble understaning this step:

    "Since 2+i is in A we have i+A=-2+A"

    and then it does it again "Since 5 is in A 5+A=0+A"

    why is this?

    thanks
     
  2. jcsd
  3. Mar 22, 2009 #2

    Office_Shredder

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    The first one is, uh, wrong in its reasoning.

    Note in general, if a-b is in an ideal I, then a+I = b+I. You should try proving this on your own. In particular, 5 = 5-0 is in A, so 5+A = 0+A
     
  4. Mar 22, 2009 #3
    thanks Shredder:

    I have no idea how to prove that ive been trying for an hour
     
  5. Mar 23, 2009 #4

    matt grime

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    I don't see why office shredder called the first piece of reasoning wrong. I'll write [a] for the coset a+I. The first statement is just

    1) [2+i]= [0] (certainly true)

    2) [2+i] + [-2] = [0] + [-2]

    3) [2+i -2] = [0 - 2]

    4) =[-2]

    All fine there.

    What part of the second bit is troubling you? Write out what you've done.
     
  6. Mar 23, 2009 #5

    Office_Shredder

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    Whoops, I thought it said i+A = 2+A. Missed the - sign there
     
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