• Beachy, John A. (Department of Mathematical Sciences Northern Illinois University) ;
  • Medina-Barcenas, Mauricio (Facultad de Ciencias Fisico Matematicas Benemerita Universidad Aut onoma de Puebla)
  • Received : 2019.09.26
  • Accepted : 2020.07.08
  • Published : 2020.09.30


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.


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