Property: linearly compact

Definition: Given any family of submodules $N_i$ of $M$ indexed by $I$, and elements $m_i$ indexed by the same set, if $\bigcap_{i\in F}(m_i+N_i)\neq \emptyset$ for every finite subset $F$ of $I$, then $\bigcap_{i\in I}(m_i+N_i)\neq \emptyset$. (In other words, every finitely-solvable system of congruence is solvable.)

Reference(s):

D. Zelinsky. Linearly compact modules and rings. (1953) @ whole article

Metaproperties:

This property has the following metaproperties

passes to submodules
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