On flatness and coherence with respect to modules of flat dimension at most one

TL;DR

This paper explores new classes of modules related to flat dimension at most one, introduces $_1^{p}$-coherent rings, and examines their properties and inclusions of well-known ring classes.

Contribution

It defines $_1$-flat and $_1^{p}$-flat modules, introduces $_1^{p}$-coherent rings, and characterizes their relationship with existing ring classes.

Findings

$_1^{p}$-coherent rings include coherent, perfect, and semi-hereditary rings.

The class of $_1^{p}$-coherent rings is large and encompasses many important ring types.

Integral domains are a notable example satisfying $igcup_{ ightarrow}_1=_1$.

Abstract

This paper introduces and studies homological properties of new classes of modules, namely, the $\mathcal F_1$-flat modules and the $\mathcal F_1^{\fp}$-flat modules, where $\mathcal F_1$ stands for the class of right modules of flat dimension at most one and $\mathcal F_1^{\fp}$ its subclass consisting of finitely presented elements. This leads us to introduce a new class of rings that we term $\mathcal F_1^{\fp}$-coherent rings as they behave nicely with respect to $\mathcal F_1^{\fp}$-flat modules as do coherent rings with respect to flat modules. The new class of $\mathcal F_1^{\fp}$-coherent rings turns out to be a large one and it includes coherent rings, perfect rings, semi-hereditary rings and all rings $R$ such that $\lim\limits_{\lr}\mathcal P_1=\mathcal F_1$. As a particular case of rings satisfying $\lim\limits_{\lr}\mathcal P_1=\mathcal F_1$ figures the important class of integral domains.