If $f,g: M \longrightarrow M$ are $R$ module homomorphisms, then $f \circ g \equiv 0 \iff g \circ f \equiv 0$?

Question

I'm working on a series of problems on Module theory, and came across the following problem:

If $f,g: M \longrightarrow M$ are $R$ module homomorphisms, then $f \circ g \equiv 0 \iff g \circ f \equiv 0$?

I suspect that this is false, but I can't think of a counter example. I tried looking at polynomial rings as modules over themselves, as well as rings with trivial module structure for an easy counter example, but can't think of anything.

Any help would be appreciated.