Bulletin of the
Korean Mathematical Society BKMS

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.

Keywords: Goldie module, prime radical, nilpotent submodule, retractable module, ACC on annihilators, projective module

MSC numbers: Primary 16P60, 16D70; Secondary 16D40, 16D80

Supported by: The second author was supported by the grant ``CONACYT-Estancias Posdoctorales 1er A\~no 2019-1''.