Extension of the Rankin–Cohen algebra proposition to arbitrary graded weights

Let $M_\ast$ be a graded algebra that may contain elements of negative weights or non-constant elements of weight zero, with

$$M_\ast=\bigoplus_{k\in\mathbb{Z}}M_k.$$

Generalized Rankin–Cohen proposition. The proposition concerning Rankin–Cohen algebras holds for such a graded algebra; it is not necessary that $M_0={\sf k}.1$. The paper identifies this extension as the difficulty underlying the conjectural closure of the space of 2CY modular forms under Rankin–Cohen brackets, and provides no proof, so the assertion remains open.