# ON AN UPPER NIL RADICAL FOR NEAR-RING MODULES

### Abstract

For a near-ring R we introduce the notion of an s−prime R−module and an s−system. We show that an R−ideal P is an s−prime R−ideal if and only if R\P is an s−system. For an R−ideal N of the near-ring module M we define S(N) =: {m ∈ M: every s−sytem containing m meets N} and prove that it coincides with the intersection of all the s−prime R−ideals of M containing N. S(0) is an upper nil radical of the near-ring module. Furthermore, we define a T −special class of nearring modules and then show that the class of s−prime modules forms a T −special class. T −special classes of s−prime near-ring modules are then used to describe the 2-s-prime radical of a near-ring.