If $0 \rightarrow M' \rightarrow M \xrightarrow[]{f} M'' \rightarrow 0$ is exact then $M$ is Noetherian iff $M',M''$ are.
7645
Noetherian property for exact sequence
122189
Show that $M[x]$ is a Noetherian $A[x]$-module.
3129
A commutative ring is a field iff the only ideals are $(0)$ and $(1)$
53548
$M' \to M \to M'' \to 0$ exact $\implies 0\to \text{Hom}(M'',N) \to \text{Hom}(M,N) \to \text{Hom}(M',N)$ is exact.
69012
How to directly prove every element in a ring with exactly three ideals is either a unit or a zero divisor?
135511
$M_a =\{ f\in C[0,1] |\ f(a)=0 \}$ for $a$ $\in$ $[0,1]$. Is $M_a$ finitely generated in $C[0,1]$?
69141
Is every Noetherian module finitely generated?
128861
If $M$ isn't Noetherian, $M$ has a submodule maximal with respect to being not finitely generated.
37479
Proper submodules of Noetherian modules are intersections of indecomposable modules
49303
In a noetherian integral domain every non invertible element is a product of irreducible elements
76661
Checking the maximality of a principal ideal in $R[x]$
158311
What is the dimension of $R[[X]]$ where $R$ is a Noetherian ring?
138469
Is a ring of integers necessarily Noetherian?
52602
$K[X^2,X^3]\subset K[X]$ is a Noetherian domain and all its prime ideals are maximal
28244
Is the ideal $I = \{f\mid f (0) = 0\}$ in the ring $C [0, 1]$ of all continuous real valued functions on $[0, 1]$ a maximal ideal?
79591
Exact Sequences of Modules and Rank
156457
Show that $K[x,xy,xy^2,\dots]$ is not Noetherian