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
$(x + (x,y)^2)$
$(x)/(x^2)$
$2$-adic integers: $\mathbb Z_2$
$\bigoplus_{i=0}^\infty \mathbb Q$
$\bigoplus_{i=1}^\infty \mathbb Z$
$\bigoplus_{i=1}^\infty \mathbb Z_{(2)}$
$\bigoplus_{i=1}^\infty F_2$
$\mathbb Q$
$\mathbb Q\times \frac{\mathbb Z}{p\mathbb Z}$, where $p$ is prime
$\mathbb Q^n$
$\mathbb R[x_1,x_2,x_3,\ldots]$
$\mathbb Z$
$\mathbb Z\times \frac{\mathbb Z}{p\mathbb Z}$, $p$ prime
$M_n(\mathbb Q)$
$T_n(F_2)$
$Z(p^\infty)$: 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