Ore meets homothetic extensions

Sufficient and necessary conditions for an extension of a skew-derivation $(\delta_R,\alpha_R)$ of an associative $\mathbb{F}$-algebra $R$ to a skew derivation $(\delta_S,\alpha_S)$ on an extension $S$ of $R$ by $\mathbb{F}$ or a {\em homothetic extension $S$ of $R$ by $\mathbb{F}$} are derived. It is then shown that this yields the unique extension of the Ore extension $R[x;\alpha_R,\delta_R]$ of $R$ by $\mathbb{F}$ that embeds in the Ore extension $S[x;\alpha_S,\delta_S]$ of $S$ by the extended skew-derivation.