Modules with few types
Let \(R\) be a countable ring. A left \(R\)-module \(M\) is said to have few types if there are just countably many complete types in a finite number of variables consistent with \(M\) (in the language of left \(R\)-modules). It is proved in this paper that if \(M\) has few types, then the pure- injective hull of a type in finitely many variables is a direct sum of finitely many indecomposable modules. It follows that \(M\) has few types iff the direct sum of infinitely many copies of \(M\) has few types. If the ring \(R\) is commutative, then the annihilator ideal of any subset of \(M\) is shown to have a primary decomposition. If \(M\) has few types, a regular generic type and is not definably connected-by-finite, it is proved that \(M\) must be weakly minimal. If the ring \(R\) is right Noetherian, we settle Vaught's Conjecture for a weakly minimal left \(R\)-module \(M\) by showing that if \(M\) has few types, then it must be totally transcendental.
scientific article; zbMATH DE number 63617