If $\varphi$ and $\psi$ are two plurisubharmonic weights on an algebraic manifold $X$, then we have $\mathcal{J}(\varphi+\psi)\subseteq \mathcal{J}(\varphi)\mathcal{J}(\psi)$.

My question is, if we only allow algebraic functions in such ideals, that is, replace $\mathcal{J}(\varphi)$ by $\mathcal{J}(\varphi) \bigcap \mathcal{O}$, where $\mathcal{O}$ is the algebraic structure sheaf of $X$. Do we still have $\mathcal{J}(\varphi+\psi)\subseteq \mathcal{J}(\varphi)\mathcal{J}(\psi)$?

Maybe it's a simple question, but I'm not familar with the analytic settings...