Abstract
In this paper, we study the left orthogonal class of max-flat modules which are the homological objects related to s-pure exact sequences of modules and module homomorphisms. Namely, a right module $A$ is called MF-projective if ${Ext}^{1}_{R}(A,B)=0$ for any max-flat right $R$-module $B$, and $A$ is called strongly MF-projective if ${Ext}^{i}_{R}(A,B)=0$ for all max-flat right $R$-modules $B$ and all $i\geq 1$. Firstly, we give some properties of $MF$-projective modules and SMF-projective modules. Then we introduce and study MF-projective dimensions for modules and rings. The relations between the introduced dimensions and other (classical) homological dimensions are discussed. We characterize some classes of rings such as perfect rings, $QF$ rings and max-hereditary rings by $(S)MF$-projective modules. We also study the rings whose right ideals are MF-projective. Finally, we characterize the rings whose $MF$-projective modules are projective.
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