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CHAIN CONDITIONS ON ÉTALE GROUPOID ALGEBRAS WITH APPLICATIONS TO LEAVITT PATH ALGEBRAS AND INVERSE SEMIGROUP ALGEBRAS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 104 / Issue 3 / June 2018
- Published online by Cambridge University Press:
- 28 March 2018, pp. 403-411
- Print publication:
- June 2018
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- Article
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The author has previously associated to each commutative ring with unit
$R$ and étale groupoid 
$\mathscr{G}$ with locally compact, Hausdorff and totally disconnected unit space an 
$R$-algebra 
$R\,\mathscr{G}$. In this paper we characterize when 
$R\,\mathscr{G}$ 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 Okniński of Noetherian inverse semigroup algebras and of Zelmanov of Artinian inverse semigroup algebras.
