Hi all!

I am interested in the following question in homological algebra.

Let we have two noncommutative rings with homomorphism $\phi:B\rightarrow A$ and $M$ be a projective $A$-module. Consider the following extension of $M$ over $B$

$0\rightarrow M\rightarrow N\rightarrow M\rightarrow0$

What is the obstruction for $N$ to be a projective $B$-module?

In other words, which elements from $\text{Ext}^{1}_{B}(M,M)$ correspond to projective modules?