6. Proposition 4.6.3 - Noetherian respects exactness

Consider an exact sequence of $A$ -modules:

 \usepackage{amssymb} 
 \usepackage{tikz-cd} 
  \begin{document} 
 \Large 
 \begin{tikzcd} 
 0 \arrow[r] & M' \arrow[r, "f"] & M\arrow[r, "g"] & M'' \arrow[r] &0 
 \end{tikzcd} 
 \end{document}

Then we have that $M$ is Noetherian/Artininian iff 1;; $M^{'}, M^{″}$ are noetherian/artinian

Remark

Recall that if $M^{″}$ and $M^{'}$ are only finitely generated, then $M$ is finitely generated. The converse is equivalent to $M$ being Noetherian.