Ore meets homothetic extensions

TL;DR

This paper establishes conditions for extending skew-derivations from an algebra to its extensions, leading to a unique embedding of Ore extensions in the context of homothetic extensions.

Contribution

It provides necessary and sufficient conditions for extending skew-derivations to homothetic extensions, ensuring a unique embedding of Ore extensions.

Findings

Derived conditions for extending skew-derivations.

Proved the uniqueness of Ore extension embeddings.

Connected skew-derivations with homothetic extensions.

Abstract

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.