View source for Cotorsion-free group

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110450/c1104501.png" /> of rational numbers, or to the additive group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110450/c1104502.png" />-adic integers for any prime <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110450/c1104503.png" />, and contains no [[Cyclic group|cyclic group]] of prime order (thus, it is torsion-free). Equivalently, an Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110450/c1104504.png" /> is cotorsion-free if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110450/c1104505.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110450/c1104506.png" /> denotes the [[Completion|completion]] of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110450/c1104507.png" /> of integers in its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110450/c1104508.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110450/c1104509.png" /> with identity is isomorphic to the endomorphism ring of a countable reduced torsion-free Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110450/c11045010.png" />. Moreover, if the additive group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110450/c11045011.png" /> has finite rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110450/c11045012.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110450/c11045013.png" /> can be chosen to have rank at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110450/c11045014.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110450/c11045015.png" /> be a cotorsion-free ring with identity and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110450/c11045016.png" /> a cardinal number such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110450/c11045017.png" />. There are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110450/c11045018.png" /> cotorsion-free groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110450/c11045019.png" /> of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110450/c11045020.png" /> whose endomorphism rings are isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110450/c11045021.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110450/c11045022.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110450/c11045023.png" />.

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110450/c11045024.png" /> 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]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110450/c11045025.png" /> with identity is isomorphic to the finitely topologized endomorphism ring of a cotorsion-free Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110450/c11045026.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110450/c11045027.png" /> is complete Hausdorff in its topology and admits a base of neighbourhoods of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110450/c11045028.png" /> consisting of left ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110450/c11045029.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110450/c11045030.png" /> 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>