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

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The discussion focuses on the definition of localization of a ring, specifically addressing the confusion around localizing at an ideal versus a prime ideal. It clarifies that when S is an ideal containing 0, the localization S^-1R becomes trivial and uninteresting. The more relevant case is localizing at a prime ideal P, where the multiplicative set S is defined as R\P, ensuring that 0 is excluded. This results in S^-1R being a local ring with a unique maximal ideal.

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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.
 
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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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