Throughout I am looking at one parameter group laws.
Let $R$ be a commutative ring with identity. Let $\mathbb{G}_a(X,Y) \in R[[X,Y]]$ be the formal additive group law, i.e., $\mathbb{G}_a(X,Y)= X+Y$.
I proved that $\operatorname{End}(\mathbb{G}_a) \cong R$, when $R$ has characteristic $0$.
I am trying to figure out what $\operatorname{End}(\mathbb{G}_a)$ will be when $R$ is a field of positive characteristic $p$.
I think I managed to show that any element of $\operatorname{End}(\mathbb{G}_a)$ in this case is of the form $a_oT + a_1T^{p^1} + a_2T^{p^2}+...$ But, I am not quite sure what the ring $\operatorname{End}(\mathbb{G}_a)$ should be.
Any help would be appreciated.
It is possible that the elements of $\operatorname{End}(\mathbb{G}_a)$ do not have the form I think they do. In that case it would be helpful to know what form the elements of $\operatorname{End}(\mathbb{G}_a)$ have.