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 $M^{(\Lambda )}$-projective for every index set $\Lambda$. This extends the well known fact that in a left Goldie ring, the prime radical is nilpotent.