The answers to this question prove that over an integral domain the functional equation $$f(x+y+z)=f(x)g(y)+f(y)g(z)+f(z)g(x)$$ has only solutions where one of $f$ or $g$ is constant. The proofs make frequent use of the unity element $1\neq 0$, the zero divisors property and a few uses of the commutativity.
Is there a solution to this equation with non-constant $f$ and non-constant $g$ over some ring that is not an integral domain? Are there examples in rings that lack just one of the properties of an integral domain (commutativity, unity element, zero divisors property)?