The dual of the socle-fine notion and applications (Q2785945)

A class of unitary left \(A\)-modules is said to be socle-fine if for every pair of modules \(M\) and \(N\) in the class, \(M\) and \(N\) are isomorphic if and only if the socle of \(M\) is isomorphic to the socle of \(N\). This notion was introduced and studied by the authors [in Lect. Notes Pure Appl. Math. 153, 171-179 (1993; Zbl 0829.16003)]. They were able to show that a ring \(A\) is semiartinian if and only if the class of injective \(A\)-modules is socle-fine. The authors followed this paper with [Commun. Algebra 23, No. 14, 5329-5338 (1995; Zbl 0842.16004)] where they proved that a ring \(A\) is semisimple if and only if the class of quasi-injective \(A\)-modules (the class of quasi-projective \(A\)-modules) is socle-fine. They also proved several other interesting results in that article. For example, they showed that \(A\) is a left noetherian \(V\)-ring if and only if the class of quasi-injective \(A\)-modules is socle-fine.NEWLINENEWLINENEWLINEIn the paper under review, the authors introduce the dual of the notion of a socle-fine class. They call a class of unitary left \(A\)-modules radical-fine if for any pair of modules \(M\) and \(N\) in the class, \(M\) and \(N\) are isomorphic if and only if \(M/\text{Rad}(M)\) is isomorphic to \(N/\text{Rad}(N)\) where \(\text{Rad}(\cdot)\) denotes the Jacobson radical of the module. They show that \(A\) is a \(V\)-ring if and only if \(A\)-Mod is radical-fine. Moreover, they use the radical-fine condition to show that a left artinian QF-3 ring \(A\) is quasi-Frobenius if and only if the class of injective \(A\)-modules is radical-fine. This leads to the dual observation that a left artinian QF-3 ring \(A\) is quasi-Frobenius if and only if the class of projective \(A\)-modules is socle-fine. Finally, they prove that the following conditions are equivalent when \(A\) is a cogenerator: 1) \(A\) is QF-3, 2) \(A\) is pseudo-Frobenius, and 3) The class of projective \(A\)-modules is socle-fine.NEWLINENEWLINEFor the entire collection see [Zbl 0855.00015].