Property: amply supplemented

Definition: For every submodule $N$ of $M$, and every submodule $T$ such that $T+N=M$, there exists a submodule $S$ contained in $T$ minimal with the property that $S+N=M$.

Reference(s):

R. Wisbauer. Foundations of module and ring theory. (2018) @ Chapter 8 , section 14, subsection 9 p 354

Metaproperties:

This property has the following metaproperties

passes to summands
passes to quotients

Modules

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