Let M be a right R-module. In this paper a generalization of the notion of an s-system of rings to modules is given. Let N be a fully invariant submodule of M. Define  every s-system containing m meets

It is shown that  is equal to the intersection of all s-prime submodules of M containing N. We define  This is called (Kothe’s) upper nil radical of M. We show that if M is a quasi-projective and finitely generated right R-module which is a self-generator, then   is the sum of all fully invariant nil submodules of M. Finally we introduce the notion of NI-modules and NI submodules.

s-prime modules, upper nil radical, nil submodules, s-systems of modules.