Exploring virtual versions of \(H\)-supplemented and \(NS\)-modules (Q6600966)

Recall that a submodule \(N\) of \(M\) is called \textit{small} in \(M\) if there does not exist any proper submodule \(K\) of \(M\) such that \(M=N+K\). A module \(M\) is said to be \textit{supplemented} if, for any submodule \(T\) of \(M\), there exists a submodule \(H\) of \(M\) such that \(M=T+H\) and \(T\cap H\) is a small submodule of \(H\). The concept of an \textit{H-supplemented module} was first introduced in [Continuous and discrete modules. Cambridge etc.: Cambridge University Press (1990; Zbl 0701.16001)] by \textit{S. H. Mohamed} and \textit{B. J. Müller}, where a module \(M\) is said to be H-supplemented if for every submodule \(N\) of \(M\), there exists a direct summand \(D\) of \(M\) such that \(M=N+X\) if and only if \(M=D+X\) for every submodule \(X\) of \(M\). Also, an \textit{NS-module} is called a module in which every noncosingular submodule is a direct summand (see [\textit{Y. Talebi} et al., Bull. Iran. Math. Soc. 43, No. 3, 911--922 (2017; Zbl 1403.16003)]).\N\N\NThe aim of this paper is to study and introduce the notion of virtual versions of \(H\)-supplemented modules and \(NS\)-modules. These modules are defined by replacing the condition of being a ``\textit{direct summand}'' with being ``\textit{isomorphic to a direct summand}''. The paper explores various equivalent conditions for a module to be virtually H-supplemented and investigates their fundamental properties.