Modules with Few Models over a Serial Ring

Vera Puninskaya

An R-module M is called uniserial if the lattice of submodules of M is a chain. A module that is a direct sum of uniserial modules is called serial. A ring R is called serial if both R_{R} and _{R}R are (serial) modules. A module (over a countable ring) is said to have few types if its theory has only countably many n-types for every natural number n. A module is said to have few models if its theory has less than continuum many countable models.

It is proved Theorem. Let M be a right module over a countable serial ring R. The following conditions are equivalent:

1. M has few types;

2. M has few models;

3. M is Σ-pure–injective.

It gives a generalisation of the previous result of the author [1] that every module with few models over a countable serial ring is Σ-pure–injective.

[1] V. A. Puninskaya, Vaught’s conjecture for modules over a serial ring, J. Symb. Logic, to appear.

Address: Chair of Algebra, Moscow State University, 119899, Moscow, Russia

E-mail: punins@orc.ru