On The Spectrum Of A Ring

Algebraic Geometry
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Tsakanikas Nickos
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On The Spectrum Of A Ring

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Post by Tsakanikas Nickos »

  1. Let \( A \) be a ring. Show that \( Spec(A) \) is not connected if and only if \( A \) is isomorphic to the product of two non-zero rings \( R \) and \( S \), if and only if \( A \) contains non-trivial idempotents.
  2. If \( A \) is an integral domain, then show that \( Spec(A) \) is irreducible.
  3. Show that \( Spec(A) \) is irreducible if and only if \( A \) has a unique minimal prime ideal, if and only if the nilradical of \( A \) is a prime ideal.
  4. Prove that the spectrum \( Spec(L) \) of a local ring \( L \) is connected.
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