Multiplicative structure on solution space of a linear PDE

Question

Given a differential operator, $L$, on $\mathbb{R}^n$. I want to show that if $L$ satisfies the following properties $$ Lf = Lg = 0 \implies L(f + g) = 0, \qquad L(fg) = 0 $$ then $$ L = \sum_{i = 1}^n c_i \frac{d}{dx^i}, \qquad c_i \in \mathbb{R} $$ I'm not sure how to rule out that higher order differential operators could have this nice multiplicative structure on the solution space.