Property: proper Jacobson radical

Definition: $M$ has a maximal submodule, so that $J(M)$, the intersection of maximal submodules of $M$, is a proper submodule.

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This property does not have the following metaproperties

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

Modules

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