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Every field is an integral domain?

  1. Jan 10, 2015 #1
    Okay, so if r,s are elements of R and rs = 0 then either r or s has to equal zero.

    I'm just confused because it seems that if rs = 0 then we can show that both r and s must be zero.... I don't know what i'm doing wrong..

    rs = 0
    r^-1(rs) = r^-1(0) = 0
    1s = 0
    s = 0

    and then:

    rs = 0
    r(ss^-1) = (0)s^-1 = 0
    r1 = 0 = 0
    r = 0

    I mean, both r and s have multiplicative inverses so it seems like r and s both need to be zero by this logic.. what am i doing wrong here?
     
  2. jcsd
  3. Jan 10, 2015 #2

    mfb

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    2016 Award

    Staff: Mentor

    Not if they are zero.

    5*0=0 in the real numbers
    Now try to apply your prove to show 5=0. It won't work.
     
  4. Jan 15, 2015 #3

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    You are assuming, when you write this, that r has an inverse. Which is only true if r is not 0.
    So what you have proved is "if r is not 0 then s is 0"

    again, you have proved that "if s is not 0 then r is 0".

    no, this is not true.
     
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