Property: semi-Noetherian

Definition: Every nonzero submodule of $M$ has a maximal submodule. Also known as: max modules, Hamsher modules

C. Faith. Rings whose modules have maximal submodules. (1995) @ (all)
J. Clark, C. Lomp, N. Vanaja, and R. Wisbauer. Lifting modules: supplements and projectivity in module theory. (2008) @ Section 2.19 p 17

This property has the following metaproperties

This property does not have the following metaproperties

passes to quotients (Counterexample: $M_{ 3 }$ is a homomorphic image of $M_{ 16 }$)

Modules

	Name
	$\bigoplus_{i=1}^\infty \mathbb Z_{(2)}$
	$\mathbb R[x_1,x_2,x_3,\ldots]$
	Interval monoid ring (right regular module)
	$\mathbb Q$
	$\mathbb Q\times \frac{\mathbb Z}{p\mathbb Z}$, where $p$ is prime
	$Z(p^\infty)$: the Prüfer $p$ group
	Finitely cogenerated, not Artinian
	Uniserial, not endolocal module
	$(x + (x,y)^2)$
	$(x)/(x^2)$
	$2$-adic integers: $\mathbb Z_2$
	$\bigoplus_{i=0}^\infty \mathbb Q$
	$\bigoplus_{i=1}^\infty \mathbb Z$
	$\bigoplus_{i=1}^\infty F_2$
	$\mathbb Q^n$
	$\mathbb Z$
	$\mathbb Z\times \frac{\mathbb Z}{p\mathbb Z}$, $p$ prime
	$M_n(\mathbb Q)$
	$T_2(F_2)$
	A 2-generated faithful torsion module
	Indecomposable, not uniform module