On $\tau_q$-flatness and $\tau_q$-coherence

TL;DR

This paper introduces and explores $ au_q$-flat modules and $ au_q$-coherent rings, establishing their properties, characterizations, and relations to existing ring concepts, with implications for finitistic dimensions and regularity.

Contribution

It defines $ au_q$-flatness and $ au_q$-coherence, proves their key properties, and establishes the Chase theorem for $ au_q$-coherent rings, advancing the theory of torsion-related ring structures.

Findings

Small finitistic dimensions of T$(R[x])$ are zero for any ring $R$.

A ring is $ au_q$-VN regular iff T$(R[x])$ is von Neumann regular.

Characterization of $ au_q$-coherent rings via direct products and flat modules.

Abstract

In this paper, we introduce and study the notions of $\tau_q$-flat modules and $\tau_q$-coheret rings. First, by investigating the Nagata rings of $\tau_q$-torsion theory, we show that the small finitistic dimensions of T$(R[x])$ are all equal to $0$ for any ring $R$. Then, we introduce the notion of $\tau_q$-VN regular rings (i.e. over which all modules are $\tau_q$-flat), and show that a ring $R$ is a $\tau_q$-VN regular ring if and only if T$(R[x])$ is a von Neumann regular ring. Finally, we obtain the Chase theorem for $\tau_q$-coheret rings: a ring $R$ is $\tau_q$-coherent if and only if any direct product of $R$ is $\tau_q$-flat if and only if any direct product of flat $R$-modules is $\tau_q$-flat. Some examples are provided to compare with the known conceptions.