On strongly flat and weakly cotorsion modules

TL;DR

This paper characterizes classes of strongly flat and weakly cotorsion modules over commutative rings, introduces related module classes, and establishes their properties over Noetherian rings with countable spectra.

Contribution

It provides new characterizations and generation procedures for strongly flat and weakly cotorsion modules, including their variants over countable multiplicative subsets.

Findings

Strongly flat modules characterized by specific conditions.

Weakly cotorsion modules generated via a new procedure.

Over Noetherian rings with countable spectrum, all flat modules are quite flat.

Abstract

The aim of this paper is to describe the classes of strongly flat and weakly cotorsion modules with respect to a multiplicative subset or a finite collection of multiplicative subsets in a commutative ring. The strongly flat modules are characterized by a set of conditions, while the weakly cotorsion modules are produced by a generation procedure. Passing to the collection of all countable multiplicative subsets, we define quite flat and almost cotorsion modules, and show that, over a Noetherian ring with countable spectrum, all flat modules are quite flat and all almost cotorsion modules are cotorsion.