Modules and Stability Theory

Abstract

Modules are now widely recognized as important examples of stable structures. In fact, in the light of results and conjectures of Zilber [Zi] (${\aleph _1}$-categorical structures are “field-like”, “module-like” or “trivial”), we may consider modules as one of the typical examples of stable structures. Our aim here is both to prove some new results in the model theory of modules and to highlight the particularly clear form of, and the algebraic content of, the concepts of stability theory when applied to modules. One of the main themes of this paper is the connection between stability-theoretic notions, such as ranks, and algebraic decomposition of models. We will usually work with $T$, a complete theory of $R$-modules, for some ring $R$. In $\S 2$ we show that the various stability-theoretic ranks, when defined, are the same. In $\S 3$ we show that $T$ (not necessarily superstable) is nonmultidimensional (in the sence of Shelah [Sh1]). In $\S 4$ we consider the algebraic content of saturation and we show, for example, that if $M$ is a superstable module then $M$ is $F_{{\aleph _0}}^a$-saturated just if $M$ is pure-injective and realizes all types in finitely many free variables over $\phi$. In $\S 5$ we use our methods to reprove Ziegler’s theorem on the possible spectrum functions. In $\S 6$ we show the profusion (in a variety of senses) of regular types. In $\S 7$ we give a structure theorem for the models of $T$ in the case where $T$ has $U$-rank 1.