A construction of relatively pure submodules

TL;DR

This paper generalizes a classical theorem on the existence of relatively pure submodules from modules over rings with a unit to modules over rings with multiple objects, with applications to alpha-pure objects.

Contribution

It extends the classical theorem to more general module categories over rings with several objects, broadening its applicability.

Findings

Generalized the existence theorem for relatively pure submodules

Applied the theorem to alpha-pure objects in complex module categories

Provided new insights into the structure of modules over rings with multiple objects

Abstract

We reconsider a classical theorem by Bican and El Bashir, which guarantees the existence of non-trivial relatively pure submodules in a module category over a ring with unit. Our aim is to generalize the theorem to module categories over rings with several objects. As an application we then consider the special case of alpha-pure objects in such module categories.