Hilbert series of modules over Lie algebroids

We consider modules $M$ over Lie algebroids ${\mathfrak{g}}_A$ which are of finite type over a local noetherian ring $A$. Using ideals $J\subset A$ such that ${\mathfrak{g}}_A\cdot J\subset J$ and the length $\ell_{{\mathfrak g}_A}(M/JM)< \infty$wecandefineinanaturalwaytheHilbertseriesof$M$withrespecttothedefiningideal$J$. This notion is in particular studied for modules over the Lie algebroid of $k$-linear derivations ${\mathfrak g}A=T{A/k}(I)$thatpreserveanideal$I\subset A$,forexamplewhen$A={\mathcal O}_n$, the ring of convergent power series. Hilbert series over Stanley-Reisner rings are also considered.