On second socles of finitely cogenerated injective modules (Q1277169)

Let \(E(T)\) denote an injective hull of the module \(T\). The main result of this paper states that if \(R\) is a ring for which \(E(T)/T\) is finitely cogenerated for every simple right \(R\)-module \(T\) then every finitely cogenerated seminoetherian right \(R\)-module is of finite length. (A module is called seminoetherian if each of its nonzero submodules contains a maximal submodule.) As a corollary, it is shown that if \(R\) is a right PF left semiartinian ring then the following are equivalent: (1) \(R\) is quasi-Frobenius, (2) \(R/\text{Soc}(R)\) is finitely cogenerated as a right \(R\)-module, (3) the second socle \(\text{Soc}_2(R)\) is finitely generated as a right \(R\)-module and \(\text{Soc}_2(R)/\text{Soc}(R)\) is an essential right \(R\)-submodule of \(R/\text{Soc}(R)\). This generalizes a result of \textit{Dinh van Huynh} and the reviewer [in Q. J. Math., Oxf. II. Ser. 45, No.~177, 13-17 (1994; Zbl 0805.16014)]. As a further corollary, it is shown that if \(R\) is a right perfect right finitely cogenerated ring with finitely cogenerated injective cogenerator \(U_R\) then the following are equivalent: (1) \(U/\text{Soc}(U)\) is finitely cogenerated, (2) \(U\) is finitely generated and \(R\) is right Artinian, (3) \(_SU_R\) defines a Morita duality, where \(S=\text{End}_R(U)\).