On Whitehead modules

Paul C. Eklof; Saharon Shelah

doi:10.1016/0021-8693(91)90321-X

On Whitehead modules

Paul C. Eklof^*
, Saharon Shelah

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

39 Scopus citations

Abstract

It is proved that it is consistent with ZFC + GCH that, for any reasonable ring R, for every R-module K there is a non-projective module M such that Ext_R¹(M, K) = 0; in particular, there are Whitehead R-modules which are not projective. This is generalized to show that it is consistent that, for certain rings R, there are Whitehead R-modules which are not the union of a continuous chain of submodules so that all quotients are small Whitehead R-modules. An application to Baer modules is also given: it is proved undecidable in ZFC + GCH whether there is a single test module for being a Baer module.

Original language	English
Pages (from-to)	492-510
Number of pages	19
Journal	Journal of Algebra
Volume	142
Issue number	2
DOIs	https://doi.org/10.1016/0021-8693(91)90321-X
State	Published - 1 Oct 1991

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abstract = "It is proved that it is consistent with ZFC + GCH that, for any reasonable ring R, for every R-module K there is a non-projective module M such that ExtR1(M, K) = 0; in particular, there are Whitehead R-modules which are not projective. This is generalized to show that it is consistent that, for certain rings R, there are Whitehead R-modules which are not the union of a continuous chain of submodules so that all quotients are small Whitehead R-modules. An application to Baer modules is also given: it is proved undecidable in ZFC + GCH whether there is a single test module for being a Baer module.",

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AB - It is proved that it is consistent with ZFC + GCH that, for any reasonable ring R, for every R-module K there is a non-projective module M such that ExtR1(M, K) = 0; in particular, there are Whitehead R-modules which are not projective. This is generalized to show that it is consistent that, for certain rings R, there are Whitehead R-modules which are not the union of a continuous chain of submodules so that all quotients are small Whitehead R-modules. An application to Baer modules is also given: it is proved undecidable in ZFC + GCH whether there is a single test module for being a Baer module.

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On Whitehead modules

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