Property: Hopfian

Definition: Every surjective endomorphism of $M_R$ is invertible

F. C. Leary. Dedekind finite objects in module categories. (1992) @ (whole article)

This property does not have the following metaproperties

passes to quotients (Counterexample: $M_{ 22 }$ is a homomorphic image of $M_{ 6 }$)

Modules

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