The rank of a flat module

Abstract

1. R is a ring with unit and module means unital right R-module. A directed system of R-modules (C, 0, D) consists of a directed set D and a function which associates with each aED an R-module Ca and, with each pair a, #E3D for which a<,f, a homomorphism 0,: Ca )C C A -*O is called the exact sequence of the system. Clearly A is the direct limit of the system. DEFINITION 1. A module K is said to be map-pure in C if K is a submodule of C and for each element k of K there is a map 0 from C to K with 0(k) =k.