Infinite Goldie dimensions

Abstract

Let R be a ring with identity and M a unital right R-module. By the finite Goldie dimension of M is meant the largest integer m (provided it exists) such that M contains the direct sum of m nonzero submodules. If no such m exists, then M is said to have infinite Goldie dimension. Until recently, not much attention has been given to the case of infinite Goldie dimensions, though the definition is immediate. The Goldie dimension of M-denoted Gd M-is the supremum 1, of all cardinals K such that M contains the direct sum of K nonzero submodules. We believe that one of the main reasons why infinite Goldie dimensions have been ignored is that suprema are difficult to handle, and there is no guarantee that a module M of Goldie dimension 1. contains the direct sum of exactly 2 nonzero submodules. Given a cardinal number K, we say K is attained in M if M contains a direct sum of K nonzero submodules. If K is not a limit cardinal, i.e., if it is of the form tc=H,+, for some ordinal CL, then K 2 Gd M is attained in M. Recall that an infinite cardinal K is called regular if ~~ < K for iE Z with IZl < K implies c K~ < K. Otherwise it is called singular. An uncountable, regular, limit cardinal is said to be inaccessible ([HJ, p. 1631 or [L, p. 1373). The reader is reminded that the existence of inaccessible cardinals cannot be proved in ZFC (Zermelo-Fraenkel set theory with Axiom of Choice added), and that in the constructible universe, there are no such cardinals. All proofs here are within the framework of ZFC. Our purpose here is to show that if Gd M is not an inaccessible cardinal, then Gd M is attained in M. Our result is best possible; for inaccessible cardinals see the remark at the end.