Torsion-free, divisible, and Mittag-Leffler modules

We study (relative) K-Mittag-Leffler modules, with emphasis on the class K of absolutely pure modules. A final goal is to describe the K-Mittag-Leffler abelian groups as those that are, modulo their torsion part, aleph_1-free, Cor.6.12. Several more general results of independent interest are derived on the way. In particular, every flat K-Mittag-Leffler module (for K as before) is Mittag-Leffler, Thm.3.9. A question about the definable subcategories generated by the divisible modules and the torsion-free modules, resp., has been left open, Quest.4.6

Strict Mittag-Leffler modules and purely generated classes

We study versions of strict Mittag-Leffler modules relativized to a class \cK (of modules), that is, \emph{strict} versions (in the technical sense of Raynaud and Gruson) of \cK-Mittag-Leffler modules, as investigated in the preceding paper, {\em Mittag-Leffler modules and definable subcategories}, in this very series (as well as the arXiv)

Mittag Leffler modules and definable subcategories

We study (relative) $\mathcal K$-Mittag-Leffler modules as was done in the author's habilitation thesis, rephrase old, unpublished results in terms of definable subcategories, and present newer ones, culminating in a characterization of countably generated $\cal K$-Mittag-Leffler modules

When every finitely generated flat module is projective

We investigate the class of rings over which every finitely generated flat right module is projective

When every projective module is a direct sum of finitely generated modules

AbstractWe characterize rings over which every projective module is a direct sum of finitely generated modules, and give various examples of rings with and without this property