Vol. 254, No. 1, 2011

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Strong submodules of almost projective modules

Gábor Braun and Jan Trlifaj

Vol. 254 (2011), No. 1, 73–87

The structure of almost projective modules can be better understood in the case when the following Condition (P) holds: The union of each countable pure chain of projective modules is projective. We prove this condition, and its generalization to pure-projective modules, for all countable rings, using the new notion of a strong submodule of the union.

However, we also show that Condition (P) fails for all Prüfer domains of finite character with uncountable spectrum, and in particular, for the polynomial ring K[x], where K is an uncountable field. One can even prescribe the Γ-invariant of the union. Our results generalize earlier work of Hill, and complement recent papers by Macías-Díaz, Fuchs, and Rangaswamy.

Almost projective module, pure chain, strong submodule, Γ-invariant, Prüfer domain
Mathematical Subject Classification 2010
Primary: 13C10, 16D40
Secondary: 03E75, 13F05, 13G05, 16P70
Received: 13 May 2011
Revised: 12 September 2011
Accepted: 27 September 2011
Published: 7 February 2012
Gábor Braun
Institut für Informatik
Universität Leipzig
PF 100920
D-04009 Leipzig
Jan Trlifaj
Department of Algebra
Charles University, Faculty of Mathematics and Physics
Sokolovská 83
186 75 Prague 8
Czech Republic