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.

References

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