The nilpotency of the prime radical of a Goldie module

Abstract

With the notion of prime submodule defined by F. Raggi et.al. we prove that the intersection of all prime submodules of a Goldie module M , is a nilpotent submodule provided that M is retractable and M (Λ) -projective for every index set Λ. This extends the well known fact that in a left Goldie ring, the prime radical is nilpotent.