While studying cohomology theories on the stable homotopy setting, I have come up with the following basic question:

Consider the additive formal group law, $F_a$, and the multiplicative formal group law, $F_m$, both defined over a ring $A$.

If $A$ is a $\mathbb{Q}$-algebra, the exponential series defines a morphism of formal group laws $$F_a \to F_m \quad , \quad x \mapsto e^x -1 $$ and I can check that any other morphism $\,f\colon F_a \to F_m\,$ is of the form $f (x) = e^{ax}-1$, for some $a\in A$.

The previous reasoning extends to the case when $A$ is an integral domain–of any characteristic–, and I am wondering what happens if there are zero divisors. That is to say:

Over an arbitrary ring, how many morphisms of formal group laws $F_a \to F_m$ are there?

I am also interested in endomorphisms of the multiplicative group law $F_m$. The obvious ones are rising to the $k^{th}$ power, for $k \in \mathbb{Z}$:

$$ F_m \to F_m \quad , \quad x \mapsto (1+x)^k - 1 \ , $$ and the analogous question would be:

Over an arbitrary ring, how many morphisms of formal group laws $F_m \to F_m$ are there?

Thanks in advance for any help.