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On generalizations of projectivity for modules over Dedekind domains

Part of: Abelian groups Theory of modules and ideals Homological algebra

Published online by Cambridge University Press:  09 April 2009

Jutta Hausen
Jutta Hausen
Affiliation:
Department of Mathematics, University of Houston, Central Campus Houston, Texas 77004, U.S.A.
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Abstract

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A module M over a ring R is κ-projective, κ a cardinal, if M is projective relative to all exact sequence of R-modules 0 → A → B → C → 0 such that C has a generating set of cardinality less than κ. A structure theorem for κ-projective modules over Dedekind domains is proven, and the κ-projectivity of M is related to properties of ExtR (M, ⊕ R). Using results of S. Chase, S. Shelah and P. Eklof, the existence of non-projective и1-projective modules is shown to undecidable, while both the Continuum Hypothesis and its denial (Plus Martin's Axiom) imply the existence of a reduced И0-projective Z-module which is not free.

MSC classification

Secondary: 13C05: Structure, classification theorems 13C10: Projective and free modules and ideals 18G15: Ext and Tor, generalizations, Künneth formula 20K20: Torsion-free groups, infinite rank 20K40: Homological and categorical methods
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 31 , Issue 2 , August 1981 , pp. 207 - 216
Copyright
Copyright © Australian Mathematical Society 1982

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