Operators coming from ring schemes

We introduce the notion of a coordinate $\mathbf{k}$-algebra scheme and the corresponding notion of a $\mathcal{B}$-operator. This class of operators includes endomorphisms and derivations of the Frobenius map, and it also generalizes the operators related to $\mathcal{D}$-rings from [15]. We classify the (coordinate) $\mathbf{k}$-algebra schemes for a perfect field $\mathbf{k}$ and we also discuss the model-theoretic properties of fields with $\mathcal{B}$-operators.