Bull. Korean Math. Soc. 57 (2020), No. 5, pp. 1177–1193 https://doi.org/10.4134/BKMS.b190864 pISSN: 1015-8634 / eISSN: 2234-3016 FULLY PRIME MODULES AND FULLY SEMIPRIME MODULES John A. Beachy and Mauricio Medina-B´ arcenas Abstract. 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. 1. Introduction It will be assumed throughout that R is an associative ring with identity, and that M is a fixed nonzero left R-module. A module X in R–Mod, the category of unital left R-modules, is said to be M -generated if there exists an R-epimorphism from a direct sum of copies of M onto X. The category σ[M ] of modules subgenerated by M is defined to be the full subcategory of R–Mod that contains all modules R X such that X is isomorphic to a submodule of an M -generated module. The reader is referred to [9] and [16] for results on the category σ[M ]. It is an abelian category, and in R–Mod it is closed under formation of homomorphic images, submodules, and direct sums. The results in this paper concern the analog in σ[M ] of the notion of a prime ideal of the ring R. We recall that a proper ideal P of R is said to be prime if AB ⊆ P implies A ⊆ P or B ⊆ P for all ideals A, B of R, and it is said to be semiprime if A 2 ⊆ P implies A ⊆ P for all ideals A of R, or, equivalently, if P is an intersection of prime ideals of R. A subfunctor τ of the identity on σ[M ] is called a preradical of σ[M ]; it is called a radical if τ (X/τ (X)) = (0) for all X in σ[M ]. If ρ and τ are preradicals of σ[M ] such that ρ(X) ⊆ τ (X) for all modules R X, then the notation ρ ≤ τ Received September 26, 2019; Accepted July 8, 2020. 2010 Mathematics Subject Classification. Primary 16S90, 16N60, 16D60. Key words and phrases. Prime submodule, fully prime module, semiprime submodule, fully semiprime module, regular module, fully idempotent module. c 2020 Korean Mathematical Society 1177