On Ring Theory (A Long One)

Groups, Rings, Domains, Modules, etc, Galois theory
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Tsakanikas Nickos
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On Ring Theory (A Long One)

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Post by Tsakanikas Nickos »

Let \( \displaystyle R \) be an associative ring with unity. Prove that the following are equivalent:
  1. \( \displaystyle R \) is a local ring.
  2. The set of non-invertible elements of \( \displaystyle R \) forms an ideal of \( \displaystyle R \).
  3. There is a proper left ideal of \( \displaystyle R \) containing all proper left ideals of \( \displaystyle R \).
  4. There is a proper right ideal of \( \displaystyle R \) containing all proper right ideals of \( \displaystyle R \).
Now, suppose that \( \displaystyle R \) is local and consider the set \( \displaystyle I \) of non-invertible elements of \( \displaystyle R \). Prove that
  1. \( \displaystyle I \) is the only maximal left ideal of \( \displaystyle R \) containing all proper left ideals of \( \displaystyle R \).
  2. \( \displaystyle I \) is the only maximal right ideal of \( \displaystyle R \) containing all proper right ideals of \( \displaystyle R \).
  3. \( \displaystyle I \) coincides with the (Jacobson) radical \( \displaystyle J \left( R \right) \) of \( \displaystyle R \).
  4. \( \displaystyle R/I = R/ J \left( R \right) \) is a division ring.
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