RINGS WHOSE MODULES HAVE MAXIMAL SUBMODULES

A ring R is a right max ring if every right module M = 0 has at least one maximal submodule. It suffices to check for maximal submodules of a single module and its submodules in order to test for a max ring; namely, any cogenerating module E of mod-R; also it suffices to check the submodules of the injective hull E(V) of each simple module V (Theorem 1). Another test is transfinite nilpotence of the radical of E in the sense that radα E = 0; equivalently, there is an ordinal α such that radα(E(V)) = 0 for each simple module V. This holds iff each radβ(E(V)) has a maximal submodule, or is zero (Theorem 2). If follows that R is right max iff every nonzero (subdirectly irreducible) quasi-injective right R-module has a maximal submodule (Theorem 3.3). We characterize a right max ring R via the endomorphism ring Λ of any injective cogenerator E of mod-R; namely, Λ/L has a minimal submodule for any left ideal L = annΛM for a submodule (or subset) M = 0 of E (Theorem 8.8). Then Λ/L0 has socle = 0 for: (1) any finitely generated left ideal L0 = Λ; (2) each annihilator left ideal L0 = Λ; and (3) each proper left ideal L0 = L+L′, where L = annΛM as above (e.g. as in (2)) and L ′ finitely generate