Endoproperties of modules and local duality

Let R be any ring and N=⊕i∈INi be a direct sum of finitely presented left R-modules Ni. Suppose that D(N) and D(Ni) are the local duals of N and Ni for each i∈I. We prove that the lattice of endosubmodules of N is anti-isomorphic to the lattices of matrix subgroups of D(N) and of M=⊕i∈ID(Ni). As consequences, N is endoartinian if and only if M (or D(N)) is endonoetherian, and N is endonoetherian if and only if M (or D(N)) is Σ-pure-injective. We obtain, in particular, that if R is a Krull–Schmidt ring, and M is an indecomposable pure-injective endonoetherian right R-module which is the source of a left almost split morphism in Mod(R), then M is endofinite. As an application, a ring R is of finite representation type if and only if every pure-injective right R-module is endonoetherian.