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

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.