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 ${\mathfrak{g}}_{A\mathrm{/}k}$. If $S^{\cdot }$ is semi-simple over ${\mathfrak g}{A/k}$weprovethatitsringofinvariants$\bar S^{\cdot}$ is notherian. When ${\mathfrak{g}}_{A\mathrm{/}k}$ is a solvable Lie algebra over $A$ we construct noetherian subalgebras of $S^{\cdot }$ from subsets of characters of ${\mathfrak g}{A/k}$\mathrm{.}Wegivesimilarresultsfornoetherianmodulesoverthepair$(S^{\cdot}, {\mathfrak g}_{A/k})$.