Basic Ring Theory - 9 (Chain Conditions)
Posted: Fri Dec 18, 2015 7:34 pm
- Let
\[ 0 \rightarrow M^{\prime} \rightarrow M \rightarrow M^{\prime \prime} \rightarrow 0 \]
be an exact sequence of \(A\)-modules. Show that- \( M \) is Noetherian iff \( M^{\prime} \) and \( M^{\prime \prime} \) are Noetherian.
- \( M \) is Artinian iff \( M^{\prime} \) and \( M^{\prime \prime} \) are Artinian.
- 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.
- 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.