  In this paper we improve recent results dealing with cellular covers of
$R$-modules. Cellular covers (sometimes called co-localizations) come up in the
context of homotopical localization of topological spaces. They are related to
idempotent cotriples, idempotent comonads or coreflectors in category theory.
  Recall that a homomorphism of $R$-modules $\pi: G\to H$ is called a {\it
cellular cover} over $H$ if $\pi$ induces an isomorphism $\pi_*:
\Hom_R(G,G)\cong \Hom_R(G,H),$ where $\pi_*(\phi)= \pi \phi$ for each $\phi \in
\Hom_R(G,G)$ (where maps are acting on the left). On the one hand, we show that
every cotorsion-free $R$-module of rank $\kappa<\Cont$ is realizable as the
kernel of some cellular cover $G\to H$ where the rank of $G$ is $3\kappa +1$
(or 3, if $\kappa=1$). The proof is based on Corner's classical idea of how to
construct torsion-free abelian groups with prescribed countable endomorphism
rings. This complements results by Buckner--Dugas \cite{BD}. On the other hand,
we prove that every cotorsion-free $R$-module $H$ that satisfies some rigid
conditions admits arbitrarily large cellular covers $G\to H$. This improves
results by Fuchs-G\"obel \cite{FG} and Farjoun-G\"obel-Segev-Shelah
\cite{FGSS07}.
