ON MAXIMAL TORSION RADICALS, IV John A. Beachy Department of Mathematical Sciences Northern Illinois University, DeKalb, IL 60115 E-mail: jbeachy@niu.edu ABSTRACT It is well-known that if R is a left Noetherian ring, then there is a bijective correspondence between minimal prime ideals of R and maximal torsion radicals of R–Mod. Using the notion of a prime M -ideal, it is shown that this correspondence can be extended to the category σ[M ] of modules subgenerated by a module M , pro- vided that M is a Noetherian quasi-projective generator in σ[M ]. Furthermore, under this hypothesis the prime M -ideals are the fully invariant submodules P of M such that M/P is semi-compressible. 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. Our goal is to study an individual module R M by studying the category σ[M ], in which the module M plays a role analogous to that of R R in R–Mod. Since R is a projective generator in R–Mod, to obtain parallel results in σ[M ] in certain cases we will need to assume a condition that holds whenever M is a quasi-projective generator in σ[M ]. 1