Two extensions of Hilbert's finiteness theorem

Let $S^{\cdot}$ be a noetherian graded algebra over a commutative $k$-algebra $A$, where $k$ is a commutative ring, and assume it is a module over a Lie algebroid ${\mathfrak g}_{A/k}$. If $S^\cdot$ is semi-simple over ${\mathfrak g}_{A/k}$ we prove that its ring of invariants $\bar S^{\cdot}$ is notherian. When ${\mathfrak g}_{A/k}$ is a solvable Lie algebra over $A$ we construct noetherian subalgebras of $S^\cdot$ from subsets of characters of ${\mathfrak g}_{A/k}$. We give similar results for noetherian modules over the pair $(S^{\cdot}, {\mathfrak g}_{A/k})$.

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