Let \(R\) be an associative ring with identity element. A right \(R\)-module \(M\) is said to be `distributive' if its lattice of submodules is distributive. G. M. Brodski proved in 1997 that \(M\) is distributive if and only if \(M\) has no subfactors of the form \(K\oplus N\), where \(K\) and \(N\) are isomorphic nonzero modules. Based on this characterization, A. V. Mikhalev and A. A. Tuganbaev extended in 1999 the concept of distributive module as follows: If \(\kappa\) is any cardinal number, then a module \(M\) is called `\(\kappa\)-distributive' if \(M\) has no subfactors that can be decomposed into a direct sum of \(\kappa\) copies of a nonzero module. Thus, the 2-distributive modules are precisely the usual distributive modules. The aim of this paper is to investigate the \(\kappa\)-distributive modules, where \(\kappa\) is an arbitrary cardinal number. The authors extend some characterizations of distributive modules to \(\kappa\)-distributive modules. The case \(\kappa=n\) is a finite ordinal is also considered. Then, applying the results obtained to the particular case \(n=2\) the authors provide new characterizations of distributive modules and rings.