# Cotorsion-free group

An Abelian group is cotorsion-free if it does not contain any non-zero 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 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 of the group $\mathbf Z$ of integers in its $\mathbf Z$- adic topology. Cotorsion-free rings are rings (cf. 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 [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 [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). The Corner–Göbel theorem [a2] states that a 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 [a3] generalizes to cotorsion-free algebras over commutative domains. The proof relies on the most useful black box principle.