Property: co-Hopfian

Definition: Every injective endomorphism of $M_R$ is invertible.

Reference(s):

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

Metaproperties:

This property does not have the following metaproperties
Modules
Name
$\bigoplus_{i=1}^\infty F_2$
$\mathbb Q\times \frac{\mathbb Z}{p\mathbb Z}$, where $p$ is prime
$\mathbb R/\mathbb Z$: the circle group
$\mathbb Z\times \frac{\mathbb Z}{p\mathbb Z}$, $p$ prime
$2$-adic integers: $\mathbb Z_2$
$\bigoplus_{i=1}^\infty \mathbb Z$
$\bigoplus_{i=1}^\infty \mathbb Z_{(2)}$
$\mathbb R[x_1,x_2,x_3,\ldots]$
$\mathbb Z$
Finitely cogenerated, not Artinian
Uniserial, not endolocal module
$(x + (x,y)^2)$
$(x)/(x^2)$
$\bigoplus_{i=0}^\infty \mathbb Q$
$\mathbb Q$
$\mathbb Q/\mathbb Z$
$\mathbb Q^n$
$M_n(\mathbb Q)$
$T_n(F_2)$
$Z(p^\infty)$: the Prüfer $p$ group
A 2-generated faithful torsion module
Indecomposable, not uniform module
Interval monoid ring (right regular module)