Modules over the endomorphism ring of a finitely generated projective module

Abstract:

Let ${P_R}$ be a projective module with trace ideal $T$. An $R$-module ${X_R}$ is $T$-accessible if $XT = X.{\text { If }}{P_R}$ is finitely generated projective and $C$ is the $R$-endomorphism ring of ${P_R}$, such that $_C{P_R}$, then for ${X_R}$, Horn ${({P_R},{X_R})_C}$ is artinian (noetherian) if and only if ${X_R}$ satisfies the minimum (maximum) condition on $T$-accessible submodules. Further, if ${X_R}$ is $T$-accessible then Hom ${({P_R},{X_R})_C}$ is finitely generated if and only if ${X_R}$ is finitely generated.