In this paper I study totally transcendental (tt) theories of modules in a general setting, and with a strong emphasis on model-theoretic (as opposed to algebraic) techniques. In particular, I am able to characterize fundamental stability-theoretic concepts such as non-forking independence, regularity and weight. Although the theorems on independence are not new, in my proofs I try to emphasize how algebraic properties follow naturally from the general properties of totally transcendental theories. Among other things, I am able to recover the existence and basic facts about compact (pure-injective) hulls as a consequence of these properties.
One of the important ideas underlying this work is that \(S_ 1(\emptyset)\), the set of 1-types over \(\emptyset\), is a natural analogue of the ideal lattice of a ring. Throughout this paper I relate the general results to a specific theory \(T^*_ \Lambda\), that of existentially closed modules over a Noetherian ring \(\Lambda\), where in fact \(S_ 1(\emptyset)\) ‘is’ the ideal lattice of \(\Lambda\). This analogy has been noted by other authors and exploited to great effect. In particular, see the work of Prest.
Using the ideas described above, I am able to prove two important theorems about tt theories of modules. First I provide an entirely model- theoretic proof of Garavaglia’s theorem that every tt module can be written uniquely as a direct sum of indecomposable modules. In this proof I use my characterization of the stability-theoretic concept of weight to avoid the use of an algebraic result, the Krull-Remak-Schmidt-Azumaya lemma. Then I prove a decomposition theorem for \(S_ 1(\emptyset)\) which, by the analogy between \(S_ 1(\emptyset)\) and the ideal lattice, is a generalization of Lesieur and Croisot’s extension of the classic Lasker-Noether normal decomposition theorem for ideals in a commutative Noetherian ring.