Accessible subcategories of modules and pathological objects

Let lambda be an infinite regular cardinal. It is proved, under the assumption of the Generalized Continuum Hypothesis, that any lambda-accessible and lambda-accessibly embedded subcategory K of a category of modules closed under direct sums gives rise to non-trivial kappa-separable modules, for arbitrarily large regular cardinals kappa >= lambda, when some modules of K are not direct sum of lambda-presentable modules (namely, those modules which are totally ordered.-directed colimits of lambda-presentable modules). In particular, it is shown that any ring R that is not left pure-semisimple has non-trivial lambda-separable left modules for arbitrarily large regular cardinals lambda. These results extend previous constructions by Corner, Griffith, Hill, Eklof, Shelah and Huisgen-Zimmermann. We point up that kappa-separable modules satisfy certain generalized Mittag-Leffler conditions.