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FULLY PRIME MODULES AND FULLY SEMIPRIME MODULES
Beachy, John A.,Medina-Barcenas, Mauricio Korean Mathematical Society 2020 대한수학회보 Vol.57 No.5
Fully prime rings (in which every proper ideal is prime) have been studied by Blair and Tsutsui, and fully semiprime rings (in which every proper ideal is semiprime) have been studied by Courter. For a given module M, we introduce the notions of a fully prime module and a fully semiprime module, and extend certain results of Blair, Tsutsui, and Courter to the category subgenerated by M. We also consider the relationship between the conditions (1) M is a fully prime (semiprime) module, and (2) the endomorphism ring of M is a fully prime (semiprime) ring.
THE NILPOTENCY OF THE PRIME RADICAL OF A GOLDIE MODULE
John A., Beachy,Mauricio, Medina-Barcenas Korean Mathematical Society 2023 대한수학회보 Vol.60 No.1
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<sup>(Λ)</sup>-projective for every index set Λ. This extends the well known fact that in a left Goldie ring the prime radical is nilpotent.