show/hide this revision's text 2 Fixed an oversight.

We can construct a new object from $O:=\text{End}(\phi)$ by looking at $\psi:O\to \mathscr{F} \{ \tau \}$ defined by $\psi_f = f$. This thing is not automatically a Drinfeld module, but it is close.

We can either relax our definition of Drinfeld modules or use an isogeny to assume that $\psi$ is a Drinfeld module (See the proof of Proposition 4.7.17 in Basic Structures of Function Field Arithmetic by Goss and Explicit Class Field Theory of Global Function Fields by Hayes).

Now, by looking at the torsion $\psi[a]$ for some $a\in A$ (we need to be a little careful here if $\psi$ is not really a Drinfeld module), we can see that the rank of $\psi$ is less than that of $\phi$ and also that $O$ is equal to $A$.

EDIT: I forgot to say that if $\text{End}(\phi)$ is not commutative you may have to look at a commutative subring, just like in Proposition 4.7.17 of Goss' book.
show/hide this revision's text 1

We can construct a new object from $O:=\text{End}(\phi)$ by looking at $\psi:O\to \mathscr{F} \{ \tau \}$ defined by $\psi_f = f$. This thing is not automatically a Drinfeld module, but it is close.

We can either relax our definition of Drinfeld modules or use an isogeny to assume that $\psi$ is a Drinfeld module (See the proof of Proposition 4.7.17 in Basic Structures of Function Field Arithmetic by Goss and Explicit Class Field Theory of Global Function Fields by Hayes).

Now, by looking at the torsion $\psi[a]$ for some $a\in A$ (we need to be a little careful here if $\psi$ is not really a Drinfeld module), we can see that the rank of $\psi$ is less than that of $\phi$ and also that $O$ is equal to $A$.