Bulletin of the
Korean Mathematical Society BKMS

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.

Keywords: Prime submodule, fully prime module, semiprime submodule, fully semiprime module, regular module, fully idempotent module

MSC numbers: Primary 16S90, 16N60, 16D60

Supported by: The research of the second author was supported by a Fulbright-Garc a Robles Scholarship while he was a Fulbright Scholar at Northern Illinois University