Covering classes, strongly flat modules, and completions

TL;DR

This paper explores the properties of strongly flat modules, introduces a new topology on modules over rings, and investigates the conditions under which certain classes of modules are covering and closed under direct limits.

Contribution

It defines a new topology on modules over rings, studies strongly flat modules in non-commutative settings, and examines the covering properties related to Enochs' Conjecture.

Findings

The topology coincides with Matlis' topology in the commutative case.

The completion of a ring in this topology is strongly flat.

If the class of strongly flat modules over a right chain domain is covering, then the domain is right invariant.

Abstract

We study some closely interrelated notions of Homological Algebra: (1) We define a topology on modules over a not-necessarily commutative ring $R$ that coincides with the $R$-topology defined by Matlis when $R$ is commutative. (2) We consider the class $ \mathcal{SF}$ of strongly flat modules when $R$ is a right Ore domain with classical right quotient ring $Q$. Strongly flat modules are flat. The completion of $R$ in its $R$-topology is a strongly flat $R$-module. (3) We consider some results related to the question whether $ \mathcal{SF}$ a covering class implies $ \mathcal{SF}$ closed under direct limit. This is a particular case of the so-called Enochs' Conjecture (whether covering classes are closed under direct limit). Some of our results concerns right chain domains. For instance, we show that if the class of strongly flat modules over a right chain domain $R$ is covering, then $R$ is right invariant. In this case, flat $R$-modules are strongly flat.