Abstract:

The main aim of this paper is to obtain a sufficient and necessary condition for existent of orderings on a module through obtaining a real prime submodule corresponding the real prime ideal of a commutative ring with identity 1. Then, the intersection of all orderings on a module (It is intitule ``The Core on a Real Modul" in this paper) are characterized.
The most important conclusion in this paper is a sufficient and necessary condition is obtained for judging a element of modul belongs to the core of modul.

Key words: Module, Orderings on a module, The core on a module, Fixed ideal about m