In this paper, we study the left orthogonal class of max-flat modules which are homological objects related to s-pure exact sequences and module homomorphisms. Namely, a right $A$ is called MF-projective if ${Ext}^{1}_{R}(A,B)=0$ for any $R$-module $B$, strongly ${Ext}^{i}_{R}(A,B)=0$ all $R$-modules $B$ $i\geq 1$. Firstly, give some properties $MF$-projective SMF-projective modules. Then intro...

We prove that pure morphisms of commutative rings are effective Adescent morphisms where A is a (COMMUTATIVE RINGS)-indexed category given by (i) finitely generated modules, or (ii) flat modules, or (iii) finitely generated flat modules, or (iv) finitely generated projective modules.