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

In summary, localization of a ring is a mathematical concept that involves creating a new ring from an existing one by inverting a multiplicative set of elements. This process can be defined at any ideal, but is more interesting when done at a prime ideal.
  • #1
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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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  • #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
 

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

What is the definition of localization of a ring?

The localization of a ring is a mathematical operation that allows us to create a new ring from an existing one by adding inverting elements.

What is the purpose of localizing a ring?

The purpose of localizing a ring is to allow us to perform operations and make statements that are not possible in the original ring.

How is localization of a ring different from localization of a module?

Localization of a ring and localization of a module are similar operations, but they differ in the kinds of elements that can be inverted. In localization of a ring, we can invert any non-zero element, while in localization of a module, we can only invert elements that are not contained in a particular subset called the annihilator.

What are the applications of localization of a ring in mathematics?

Localization of a ring has many applications in algebraic geometry, commutative algebra, and number theory. It is also used in the study of sheaves and schemes in algebraic geometry.

Are there any limitations to localization of a ring?

Yes, there are some limitations to localization of a ring. For example, not all rings can be localized, and the process may not always result in a commutative ring. Additionally, localization may not preserve certain properties of the original ring, such as being a finite or Noetherian ring.

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