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Math Help - Principal ideal problem

  1. #1
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    Principal ideal problem

    Let I be a principal ideal of a domain R, say I = aR. Show that I^{-1} = a^{-1}R and that I is invertible.
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  2. #2
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    Quote Originally Posted by tttcomrader View Post
    Let I be a principal ideal of a domain R, say I = aR. Show that I^{-1} = a^{-1}R and that I is invertible.
    Define what it means I^{-1}.
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  3. #3
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    The inverse of I is the set of I^{-1} = \{ u \in K : uI \subseteq R \} , where K is the quotient field of R.

    I think I have the solution:

    pick j to be an element of I^{-1}, then jI \subset R , so we have ji=r for some r in R, and i in I. Let i = am, for some elements m in R.

    So we have j(am) = r

    j(am)=r
    j=rm^{-1}a^{-1}
    j=a^{-1}(rm^{-1})

    rm^{-1} is an element of R, so I^{-1} \subseteq a^{-1}R

    Well, now I have to show that a^{-1}R is a subseteq of of I^{-1}...
    Last edited by tttcomrader; March 23rd 2008 at 10:39 AM.
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