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An [[Abelian group|Abelian group]] is cotorsion-free if it does not contain any non-zero [[Cotorsion group|cotorsion group]]. More explicitly, this means that it contains no subgroup isomorphic to the additive group  $  \mathbf Q $
of rational numbers, or to the additive group of  $  p $-
adic integers for any prime  $  p $, 
and contains no [[Cyclic group|cyclic group]] of prime order (thus, it is torsion-free). Equivalently, an Abelian group  $  A $
is cotorsion-free if and only if  $  { \mathop{\rm Hom} } ( {\widetilde{\mathbf Z}  } ,A ) = 0 $, 
where  $  {\widetilde{\mathbf Z}  } $
denotes the [[Completion|completion]] of the group  $  \mathbf Z $
of integers in its  $  \mathbf Z $-
adic topology. Cotorsion-free rings are rings (cf. [[Ring|Ring]]) whose additive groups are cotorsion-free Abelian groups. These rings play a distinguished role in the realization of rings as endomorphism rings of Abelian groups.

The celebrated theorem of Corner [[#References|[a1]]] states that any countable cotorsion-free ring  $  R $
with identity is isomorphic to the endomorphism ring of a countable reduced torsion-free Abelian group  $  A $. 
Moreover, if the additive group of  $  R $
has finite rank  $  n $, 
then  $  A $
can be chosen to have rank at most  $  2n $. 
Corner's theorem has been generalized in various directions, notably to rings of arbitrary cardinality. The best result is due to R. Göbel and S. Shelah [[#References|[a3]]]: Let  $  R $
be a cotorsion-free ring with identity and  $  \lambda $
a cardinal number such that  $  \lambda = \lambda ^ {\aleph _ {0} } \geq  | R | $. 
There are  $  2  ^  \lambda  $
cotorsion-free groups  $  A _ {i} $
of cardinality  $  \lambda $
whose endomorphism rings are isomorphic to  $  R $
such that  $  { \mathop{\rm Hom} } ( A _ {i} , A _ {j} ) = 0 $
for  $  i \neq j $.

These results have several applications to the construction of counterexamples for torsion-free Abelian groups, e.g. to Kaplansky's test problems. Another consequence is the existence of arbitrarily large indecomposable Abelian groups. There is a topological version of the above theorem. If the endomorphism ring of an Abelian group  $  A $
is equipped with the finite topology, then it becomes a complete Hausdorff topological ring (cf. also [[Hausdorff space|Hausdorff space]]). The Corner–Göbel theorem [[#References|[a2]]] states that a [[Topological ring|topological ring]]  $  R $
with identity is isomorphic to the finitely topologized endomorphism ring of a cotorsion-free Abelian group  $  A $
if and only if  $  R $
is complete Hausdorff in its topology and admits a base of neighbourhoods of  $  0 $
consisting of left ideals  $  N $
such that  $  A/N $
is cotorsion-free (the endomorphisms act on the left). The Göbel–Shelah theorem [[#References|[a3]]] generalizes to cotorsion-free algebras over commutative domains. The proof relies on the most useful black box principle.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.L.S. Corner,   "Every countable reduced torsion-free ring is an endomorphism ring"  ''Proc. London Math. Soc.'' , '''13'''  (1963)  pp. 687–710</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.L.S. Corner,   R. Göbel,   "Prescribing endomorphism algebras, a unified treatment"  ''Proc. London Math. Soc.'' , '''50'''  (1985)  pp. 447–479</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R. Göbel,   S. Shelah,   "Modules over arbitrary domains"  ''Math. Z.'' , '''188'''  (1985)  pp. 325–337</TD></TR></table>