Let $f$, $g$ be two non-constant polynomials in $\mathbb{Q}[x]$. Let $R = \mathbb{Q}[x]/(fg)$ and consider $M = \mathbb{Q}[x]/(f)$ and $N = \mathbb{Q}[x]/(g)$ as $R$-modules.
(a) Find a projective resolution of $R$-modules for $M$.
(b) Compute $\operatorname{Tor}^R_1(M,N)$.
Any help/hint in this regards would be highly appreciated
Notice that
(i) the kernel of the map $R \to fR$ is $gR$ and
(i) the kernel of the map $R \to gR$ is $fR$.
Then it is easy to show that $$ \cdots R \stackrel{\cdot g}\to R \stackrel{\cdot f}\to R \to \cdots R \stackrel{\cdot g}\to R \stackrel{\cdot f}\to R \to R/(f) \to 0 $$ is a free resolution of $M = R/(f)$.
To compute $\operatorname{Tor}^R_1(M,N)$, one may tensor the above resolution by $N$ and take the homology. Also, it is well-known that for ideals $I,J$ of $R$, $\operatorname{Tor}^R_1(R/I,R/J) = I \cap J / IJ$. Take $I=(f)$ and $J = (g)$.