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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.