# Over derivations with an inusual property

Let $$A$$ be a ring finitely generated over $$\mathbb{Z}$$. Let $$D$$ be a derivation on $$A$$. Let $$D^p$$ be the composition of $$D$$ with itself, $$p$$ times. We suppose that $$D^p(x)$$ belongs to the ideal $$pA$$, for every prime number $$p$$. Moreover, we suppose that there exist $$T$$ in $$A$$ such that $$D(T)=1$$.

Let $$B$$ be a subring of $$A$$ such that $$B$$ contains $$T$$. Let $$K(B)$$ be the field of fractions of $$B$$. We suppose that $$K(A)|K(B)$$ is algebraic. My question is if $$D$$ becomes an operator over $$K(B)$$ for a arbitrary $$B$$ with this property. It is $$D(K(B))\subset K(B)$$. Is posible that there exist a derivation whit the property of my question?

I do that question because if $$K$$ is a field of function of some component of $$A/pA$$. Let $$L$$ be the space of solutions of $$D(x)=0$$ in characteristic $$p$$. Then the condition over $$D$$ imply that $$K|L$$ is a purelly inseparable and $$K=L(T)$$. So if $$K_1$$ is such that $$K|K_1$$ is separable then $$K_1=(L\cap K_1)(T)$$. Therefore $$D(K_1)$$ is in $$K_1$$. Using the Nakayama's lemma $$D(K_1)\subset K_1$$ for every field such that $$K(A)|K_1$$ be algebraic and $$T$$ belongs to $$K_1$$. Therefore I have a derivation with that property.