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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 <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.
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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" />.
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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.
+
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====
 
====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>
 
<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>

Latest revision as of 17:31, 5 June 2020


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
How to Cite This Entry:
Cotorsion-free group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cotorsion-free_group&oldid=18619
This article was adapted from an original article by L. Fuchs (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article