On essential extensions of direct sums of either injective or projective modules. (Q2874702)

It is well known that a ring \(R\) is right Noetherian if and only if every direct sum of injective right \(R\)-modules is injective. \textit{K. I. Beidar} and \textit{W.-F. Ke} [Arch. Math. 78, No. 2, 120-123 (2002; Zbl 1020.16001)] generalized it and showed that \(R\) is right Noetherian if and only if every essential extension of a direct sum of injective hulls of simple right \(R\)-modules is a direct sum of injectives. This was further extended by \textit{P. A. Guil Asensio, S. K. Jain} and \textit{A. K. Srivastava} who showed that a ring \(R\) is right Noetherian if and only if for each injective right \(R\)-module \(M\), any essential extension of \(M^{\aleph_0}\) is a direct sum of modules that are injective or projective [J. Algebra 324, No. 6, 1429-1434 (2010; Zbl 1217.16006)].NEWLINENEWLINE The authors of this paper show that a ring \(R\) is right Noetherian if and only if every essential extension of a direct sum of injective hulls of simple right \(R\)-modules is a direct sum of either injective right \(R\)-modules or projective right \(R\)-modules.