Basic Ring Theory - 9 (Chain Conditions)

Groups, Rings, Domains, Modules, etc, Galois theory
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Tsakanikas Nickos
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Basic Ring Theory - 9 (Chain Conditions)

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Post by Tsakanikas Nickos »

  1. Let
    \[ 0 \rightarrow M^{\prime} \rightarrow M \rightarrow M^{\prime \prime} \rightarrow 0 \]
    be an exact sequence of \(A\)-modules. Show that
    1. \( M \) is Noetherian iff \( M^{\prime} \) and \( M^{\prime \prime} \) are Noetherian.
    2. \( M \) is Artinian iff \( M^{\prime} \) and \( M^{\prime \prime} \) are Artinian.
  2. Let \( A \) be a ring in which the zero ideal is a product of (not necessarily distinct) maximal ideals. Show that \( A \) is Noetherian iff \( A \) is Artinian. Give an example of such a ring.
  3. Let \( M \) be an \( A \)-module. Show that if every non-empty set of finitely generated submodules of \( M \) has a maximal element, then \( M \) is Noetherian.
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