From regular modules to von Neumann regular rings via coordinatization

TL;DR

This paper establishes a connection between regular modules and von Neumann regular rings by showing that the lattice of finitely generated submodules of a regular module can be represented as the lattice of principal right ideals of a von Neumann regular ring.

Contribution

It introduces a coordinatization method linking regular modules to von Neumann regular rings, extending Zelmanowitz's regular modules framework.

Findings

The lattice of finitely generated submodules can be coordinatized as principal right ideals.

The approach applies to modules over arbitrary rings.

Provides a new perspective on the structure of regular modules.

Abstract

In this paper we establish a very close link (in terms of von Neumann's coordinatization) between regular modules introduced by Zelmanowitz, on one hand, and von Neumann regular rings, on the other hand: we prove that the lattice $\mathcal{L}^{fg}(M)$ of all finitely generated submodules of a finitely generated regular module $M$, over an arbitrary ring, can be coordinatized as the lattice of all principal right ideals of some von Neumann regular ring $S$.