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Whitehead modules over large principal ideal domains

Whitehead modules over large principal ideal domains Abstract. We consider the Whitehead problem for principal ideal domains of large size. It is proved, in ZFC, that some p.i.d.'s of size d2 have non-free Whitehead modules even though they are not complete discrete valuation rings. 2000 Mathematics Subject Classi®cation: 03E75, 13D07. 1 A module M is a Whitehead module if ExtR MY R 0. The second author proved that the problem of whether every Whitehead Z-module is free is independent of ZFC GCH (cf. [5], [6], [7]). This was extended in [1] to modules over principal ideal domains of cardinality at most d1 . Here we consider the Whitehead problem for modules over principal ideal domains (p.i.d.'s) of cardinality b d1 . If R is any p.i.d. which is not a complete discrete valuation ring, then an R-module of countable rank is Whitehead if and only if it is free (cf. [3]). On the other hand, if R is a complete discrete valuation ring, then it is cotorsion and hence every torsion-free R-module is a Whitehead module (cf. [2, XII.1.17]). It will be convenient to decree that a ®eld is not a p.i.d. and to use the term ``slender'' to designate a p.i.d. which is not http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Whitehead modules over large principal ideal domains

Forum Mathematicum , Volume 14 (3) – Apr 15, 2002

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References (12)

Publisher
de Gruyter
Copyright
Copyright © 2002 by Walter de Gruyter GmbH & Co. KG
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/form.2002.021
Publisher site
See Article on Publisher Site

Abstract

Abstract. We consider the Whitehead problem for principal ideal domains of large size. It is proved, in ZFC, that some p.i.d.'s of size d2 have non-free Whitehead modules even though they are not complete discrete valuation rings. 2000 Mathematics Subject Classi®cation: 03E75, 13D07. 1 A module M is a Whitehead module if ExtR MY R 0. The second author proved that the problem of whether every Whitehead Z-module is free is independent of ZFC GCH (cf. [5], [6], [7]). This was extended in [1] to modules over principal ideal domains of cardinality at most d1 . Here we consider the Whitehead problem for modules over principal ideal domains (p.i.d.'s) of cardinality b d1 . If R is any p.i.d. which is not a complete discrete valuation ring, then an R-module of countable rank is Whitehead if and only if it is free (cf. [3]). On the other hand, if R is a complete discrete valuation ring, then it is cotorsion and hence every torsion-free R-module is a Whitehead module (cf. [2, XII.1.17]). It will be convenient to decree that a ®eld is not a p.i.d. and to use the term ``slender'' to designate a p.i.d. which is not

Journal

Forum Mathematicumde Gruyter

Published: Apr 15, 2002

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