Introduction

For an abelian group B, a subgroup A ⊂ B is pure in B if whenever an equation nx = c with c ∈ A and n ∈ Z has a solution x = b ∈ B in the big group B, then it already has a solution x = a ∈ A in the small subgroup A. Equivalently, A ⊂ B is pure if A ∩ nB = nA for all n ∈ Z. Direct summands of B are pure in B, and purity is a generalization of a direct summand. This chapter generalizes purity to a module context over an arbitrary ring R, where 1 ∈ R throughout this chapter.

In Section 1, an introduction to systems of equations over a module is given, which is a topic of independent interest and usefulness.

An exact sequence of modules 0 → A → B → C → 0 is pure exact if the image of A is pure in B, and a module is pure projective, by definition, if it has projective property relative to all pure exact sequences. This chapter studies pure projective modules and pure exact sequences.

Suppose that 0 → A → B → C → 0 is a short exact sequence of modules and we tensor it with a left R-module U to obtain 0 → A ⊗ U → B ⊗ U → C ⊗ U → 0. In the last chapter we saw that those modules U for which the last sequence is always exact were the flat modules. Here we ask the following converse question. Which short exact sequences have the property that the tensored sequence remains exact for every left R-module U?