Strong submodules of almost projective modules

Gábor Braun and Jan Trlifaj

Vol. 254 (2011), No. 1, 73–87

DOI: 10.2140/pjm.2011.254.73

Abstract

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.

Keywords

Almost projective module, pure chain, strong submodule, Γ-invariant, Prüfer domain

Mathematical Subject Classification 2010

Primary: 13C10, 16D40

Secondary: 03E75, 13F05, 13G05, 16P70

Milestones

Received: 13 May 2011

Revised: 12 September 2011

Accepted: 27 September 2011

Published: 7 February 2012