This thesis is based on the works on the paper ”Kothe’s upper nil radical for modules” by N. J. Groenwald and D. Ssevviiri in Acta Math. Hungar, 138(4) (2013), 295-306. Let M be a left R-module. A proper submodule P of M is called prime if for each ideal A of R and each submodule N of M if AN , then N or AM . A proper submodule P of M is called a type="#_x0000_t75" P, then AN BN and also a proper submodule P of M is called s-prime if for each ideal A of R and for each submodule N of M and for any elements x there exsites a nonzero positive integer n such that N then N or AM . In this paper generalizations of the notions of an s-system and m-system of rings to modules are given.