  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.
