Torsion free covering modules

Abstract

Let A be an integral domain and K its field of fractions. An Amodule E is said to be torsion free if ax = 0 for azA, x EE implies a = 0 or x = 0. We will say that a submodule E1 of an A -module E is pure in E if axE,=acEC\E1 for all aGA. Then if E is torsion free, a submodule E1 of E is pure in E if and only if E/E, is torsion free. Clearly the union of a chain of pure submodules of a module is still a pure submodule and if E2CE1, are submodules of E such that E2 is pure in E1 and E1/E2 pure in E/E2 then El is pure in E. It is well known that for any A-module E there exists a torsion free A -module E1 and an epimorphism p: E-*E1 such that if 4 is any linear mapping from E into a torsion free module F then there is a unique linear mapping f: El-*F such that f o p =4, i.e., the diagram