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Math Help - Associates

  1. #1
    Super Member Deadstar's Avatar
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    Associates

    R is an Integral Domain.
    Let c be a greatest common divisor of a and b in R and let f be an associate of c. Show that f is also a greatest common divisor of a and b in R.

    Thoughts...
    I think i should show that c|f and f|c. Then f would be the gcd as well.
    f = uc, where u is a unit, so c|f. Can find another unit v \in R s.t. uv. = 1 (i.e. v is the inverse of u).
    Then c = vf, hence f|c. So f is gcd as well?

    On second thoughts, since R is an Int Domain maybe im not allowed to assume that uv = 1...
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  2. #2
    Senior Member TheAbstractionist's Avatar
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    Quote Originally Posted by Deadstar View Post
    R is an Integral Domain.
    Let c be a greatest common divisor of a and b in R and let f be an associate of c. Show that f is also a greatest common divisor of a and b in R.

    Thoughts...
    I think i should show that c|f and f|c. Then f would be the gcd as well.
    f = uc, where u is a unit, so c|f. Can find another unit v \in R s.t. uv. = 1 (i.e. v is the inverse of u).
    Then c = vf, hence f|c. So f is gcd as well?

    On second thoughts, since R is an Int Domain maybe im not allowed to assume that uv = 1...
    Hi Deadstar.

    If f=uc, suppose uu'=u'u=1 for some u'. Now a=a'c and b=b'c for some a',b'. Hence a=a'u'f and b=b'u'f, showing that f divides both a and b. Now if g divides both a and b then g divides c as c is a GCD. So c=g'g for some g'. It follows that f=ug'g and so g divides f. This shows that f is a GCD.
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