Some model theory of modules. I. On total transcendence of modules

Abstract

This is the first part of a series of papers on certain topics of model theory of modules, which will further consist of Part II. On stability and categoricity of flat modules and Part III. On infiniteness of sets definable in modules. Although these parts are only weakly connected, it is convenient to have common background and notation. The first part was written in Summer 1980, when I did not know about either footnote (3) appearing in the final version of [GA 2] or about the work of the “Bedford College Group” concerning stability theory of modules. I would like to thank Wilfrid Hodges and Mike Prest for their interest in this note, which encouraged me to include it as the first part of the present series, and for acquainting me with their unpublished work, to which I refer in footnotes. I should also like to thank the referee for his suggestions, in particular for correcting the argument in Remark (5) and for drawing my attention to [Z-H].A syntactical analysis of Zimmermann's notion of “endlich matriziellen Untergruppen” of modules [ZI, p. 1087] shows that it coincides with the notion of subgroups definable by positive primitive formulae (without parameters) [GA 1, p. 80]. Then combination of [ZI, Folgerung 3.4] with [GA 1, Lemma 5] (or [GA 2, Theorem 1] for uncountable rings) yields purely algebraic criteria for total transcendence of modules. I present here a proof of this result using only the following facts.(G1) A module M is totally transcendental iff it satisfies the minimal condition on subgroups definable by positive primitive formulae (without parameters) [GA 1, Lemma 5] (and [GA 2, Theorem 1] for uncountable rings).