When it comes to the $\mathbb{F}$-vector space $\mathbb{F}^n$, it's very natural to assume $\mathbb{F}$ acts by scalar multiplication. But if the set Hom$_\textbf{Ring}(\mathbb{F})$ has more than one element, $\mathbb{F}$ has multiple ways to act on $\mathbb{F}^n$. For example, the finite field $\mathbb{F}_{25}$. Consider $\mathbb{F}_{25}$ as a 1-dimensional $\mathbb{F}_{25}$ vector space. Then the following, essentially pre-composing the usual scalar multiplication with a Frobenius map, also gives a vector space structure: $$a\cdot x:=a^5x$$ My confusion is that I haven't really heard of people talking about other vector space structures than the usual one. Do they have any importance mathematically?
There is a similar question here. My question is more of reference-requesting. Any guidance will be appreciated.
