Rings whose modules have maximal submodules

A ring $R$ is a right max ring if every right module $M\neq 0$ 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 $\operatorname{mod}$-$R$; 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 $\operatorname{rad}^{\alpha}E=0$; equivalently, there is an ordinal $\alpha$ such that $\operatorname{rad}^{\alpha}(E(V))=0$ for each simple module $V$. This holds iff each $\operatorname{rad}^{\beta}(E(V))$ 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 $\Lambda$ of any injective cogenerator $E$ of $\operatorname{mod}$-$R$; namely, $\Lambda/L$ has a minimal submodule for any left ideal $L=\operatorname{ann}_{\Lambda}M$ for a submodule (or subset) $M\ne 0$ of $E$ (Theorem 8.8). Then $\Lambda/L_0$ has socle $\ne 0$ for: (1) any finitely generated left ideal $L_0\ne\Lambda$; (2) each annihilator left ideal $L_0\ne \Lambda$; and (3) each proper left ideal $L_0=L+L'$, where $L=\operatorname{ann}_{\Lambda}M$ as above (e.g. as in (2)) and $L'$ finitely generated (Corollary 8.9A).