Operators coming from ring schemes

Abstract

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.