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I was searching for the definition of localization of a ring .

  1. Mar 9, 2008 #1
    I was searching for the definition of localization of a ring .
    I came across the definition given at

    http://mathworld.wolfram.com/Localization.html

    If i take S as an ideal, the requirement 1€S make S=R.
    I am confused here
    how can i define localization of a ring at an ideal.
     
  2. jcsd
  3. Mar 9, 2008 #2
    that link specifically says 0 is not in S, every ideal contains 0, so you cannot, if you do allow 0 to be in your multiplicative set S, then S^-1R = {0} and this is not interesting

    The more interesting case is when P is a prime ideal. Now consider the set S = R\P which is multiplicative because P is prime. It does not contain 0, and we call this localizing at the prime ideal P(even though the actual set S is R\P). In this case S^-1R is a local ring(it contains a unique maximal ideal). Maybe this is what you were thinking about
     
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