Chain conditions on \'etale groupoid algebras with applications to Leavitt path algebras and inverse semigroup algebras

The author has previously associated to each commutative ring with unit $R$ and 'etale groupoid $\mathcal{G}$ with locally compact, Hausdorff and totally disconnected unit space an $R$-algebra $R\mathcal{G}$. In this paper we characterize when $R\mathcal{G}$ is Noetherian and when it is Artinian. As corollaries, we extend the characterization of Abrams, Aranda~Pino and Siles~Molina of finite dimensional and of Noetherian Leavitt path algebras over a field to arbitrary commutative coefficient rings and we recover the characterization of Okni'nski of Noetherian inverse semigroup algebras and of Zelmanov of Artinian inverse semigroup algebras.