# Difference between revisions of "Cotorsion-free group"

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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 | + | 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