Endomorphism rings of completely pure-injective modules

Abstract:

Let $R$ be a ring, $E=E(R_R)$ its injective envelope, $S= \operatorname {End}(E_R)$ and $J$ the Jacobson radical of $S$. It is shown that if every finitely generated submodule of $E$ embeds in a finitely presented module of projective dimension $\le 1$, then every finitley generated right $S/J$-module $X$ is canonically isomorphic to $\operatorname {Hom}_R(E,X\otimes _S E)$. This fact, together with a well-known theorem of Osofsky, allows us to prove that if, moreover, $E/JE$ is completely pure-injective (a property that holds, for example, when the right pure global dimension of $R$ is $\le 1$ and hence when $R$ is a countable ring), then $S$ is semiperfect and $R_R$ is finite-dimensional. We obtain several applications and a characterization of right hereditary right noetherian rings.