This question might be trivial but I cann't see.
Let $A$ and $B$ be two modules. is it always possible to have an exact sequence which begins with $A$, ends with $B$ with all modules in the sequence (other than $A$ and $B$) projective !?
This question might be trivial but I cann't see. Let $A$ and $B$ be two modules. is it always possible to have an exact sequence which begins with $A$, ends with $B$ with all modules in the sequence (other than $A$ and $B$) projective !? 


Not in general. The keyword is stable module category, the quotient of the module category by the ideal of morphisms which factor through a projective. The leftmost term is functorial on the rightmost term in this category if all intermediate modules are projective. This imposes some restrictions. If you take a hereditary ring, eg the integers, you get easy counterexamples as any submodule of a projective module is projective. 

