A ring $R$ is called a right $\mathrm{PS}$-ring if its socle, $\mathrm{Soc}\left(R_R\right)$, is projective. Nicholson and Watters have shown that if $R$ is a right $\mathrm{PS}$-ring, then so are the polynomial ring $R\left[x\right]$ and power series ring $R\left[\right[x\left]\right]$. In this paper, it is proved that, under suitable conditions, if $R$ has a (flat) projective socle, then so does the skew inverse power series ring $R\left[[x^{-1};\alpha ,\delta ]\right]$ and the skew polynomial ring $R[x;\alpha ,\delta ]$, where $R$ is an associative ring equipped with an automorphism $\alpha$ and an $\alpha$-derivation $\delta$. Our results extend and unify many existing results....