Let $R$ be (not necessarily commutative) ring and $S$ a simple right $R$-module. Let $f\in Ann(S)$ be normalizng and a non-zero divisor. Is it always true that 
$$
pdim\_{R}(S)=pdim\_{R/(f)}(S)+1?
$$



Projective dimension over hypersurface
With the definition of normalizing you give, it is not always the case that the projective dimension of $S$ as an $R$-module equals $1$ more than the projective dimension of $S$ as an $R/\langle f \rangle$-module. Let $R$ be $\mathbb{Z}$, let $f$ be $p^2$ for some prime $p$, and let $S$ be $\mathbb{Z}/p\mathbb{Z}$. The projective dimension of $S$ as an $R$-module is $1$, but the projective dimension of $S$ as an $R/\langle f \rangle$-module is infinite.


