EKFN-modules (Q6559879)

Recall that a module \(M\) is called \textit{endo-noetherian} if, for any family (\(f_i\))\(_{i\geqslant 1}\) of endomorphisms of \(M\), the sequence \(\{\mathrm{Ker}(f_{1})\subseteq\mathrm{Ker}(f_{2})\subseteq \cdots\}\) stabilizes. The concept of \textit{EKFN-rings} was introduced by \textit{M. A. Ndiaye} and \textit{C. T. Gueye}, Int. J. Pure Appl. Math. 86, No. 5, 871--881 (2013; \url{doi:0.12732/ijpam.v86i5.10})]. A ring \(R\) is called an \textit{EKFN-ring} if every endo-Noetherian \(R\)-module is Noetherian.\N\N\N\NIn this paper, \(R\) denotes a commutative, associative ring with an identity \(1\neq 0\), satisfying the ascending chain condition (ACC) on annihilators. The aim of this paper is to extend the notion of EKFN-rings to the \(\sigma\) category (i.e., the full subcategory of \(R\)-MOD, whose objects are isomorphic to a submodule of an \(M\)-generated module). After presenting some properties, they demonstrate, under certain hypothesis, that if \(M\) is an EKFN-module, then the following equivalences hold:\N\begin{itemize}\N\item[(1)] the class of uniserial modules coincides with the class of cu-uniserial modules;\N\item[(2)] EKFN-modules correspond to the class of locally Noetherian (i.e., if every finitely generated submodule is Noetherian) modules;\N\item[(3)] the class of CD-modules is a subset of the EKFN-modules.\N\end{itemize}