Operators coming from ring schemes

We introduce the notion of a coordinate k${\bf k}$-algebra scheme and the corresponding notion of a B$\mathcal {B}$-operator. This class of operators includes endomorphisms and derivations of the Frobenius map, and it generalizes the operators related to D$\mathcal {D}$-rings from Moosa and Scanlon (J. Math. Log. 14 (2014), no. 02, 1450009) as well. We classify the (coordinate) k${\bf k}$-algebra schemes for a perfect field k${\bf k}$ and we also discuss the model-theoretic properties of fields with B$\mathcal {B}$-operators.