Let $R$ be a unital ring not necessarily commutative and $f:<a>\to<b>$ is a $R$-homomorphism of cyclic $R$-modules.
Let $X=\lbrace (r,s)\in R^2\text{ such that } r.a\neq 0 \text{ and} \quad r.f(a)+s.b=0\rbrace$.
My question is where I can know more about the homomorphisms $f$ in which $S$ is projective.