Property: has a projective cover

Definition: $M$ has a projective cover when there exists a projective module $P$ and surjective homomorphism $f:P\to M$ such that the kernel of $f$ is a small submodule of $P$.

Reference(s):

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Metaproperties:

This property does not have the following metaproperties
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
$M_n(\mathbb Q)$
$T_2(F_2)$
$Z(p^\infty)$: 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