Bulletin of the
Korean Mathematical Society

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016



Bull. Korean Math. Soc. 2023; 60(1): 185-201

Online first article January 25, 2023      Printed January 31, 2023


Copyright © The Korean Mathematical Society.

The nilpotency of the prime radical of a Goldie module

John A. Beachy, Mauricio Medina-Bárcenas

Northern Illinois University; Av. San Claudio y 18 Sur, Col. San Manuel, Ciudad Universitaria, 72570


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''.