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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.



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[1] Abrams, G., ‘Leavitt path algebras: the first decade’, Bull. Math. Sci. 5(1) (2015), 59120.
[2] Abrams, G., Ara, P. and Siles Molina, M., Leavitt Path Algebras, Lecture Notes in Mathematics, 2191 (Springer, London, 2017).
[3] Abrams, G. and Aranda Pino, G., ‘The Leavitt path algebra of a graph’, J. Algebra 293(2) (2005), 319334.
[4] Abrams, G., Aranda Pino, G. and Siles Molina, M., ‘Finite-dimensional Leavitt path algebras’, J. Pure Appl. Algebra 209(3) (2007), 753762.
[5] Abrams, G., Aranda Pino, G. and Siles Molina, M., ‘Locally finite Leavitt path algebras’, Israel J. Math. 165 (2008), 329348.
[6] Ara, P., Bosa, J., Hazrat, R. and Sims, A., ‘Reconstruction of graded groupoids from graded Steinberg algebras’, Forum Math. 29(5) (2017), 10231037.
[7] Ara, P., Hazrat, R., Li, H. and Sims, A., ‘Graded Steinberg algebras and their representations’, Preprint, 2017, arXiv e-prints.
[8] Ara, P., Moreno, M. A. and Pardo, E., ‘Nonstable K-theory for graph algebras’, Algebr. Represent. Theory 10(2) (2007), 157178.
[9] Beuter, V. and Gonçalves, D., ‘The interplay between Steinberg algebras and partial skew rings’, J. Algebra 497 (2018), 337362.
[10] Beuter, V., Gonçalves, D., Öinert, J. and Royer, D., ‘Simplicity of skew inverse semigroup rings with an application to Steinberg algebras’, Preprint, 2017, arXiv e-prints.
[11] Brown, J., Clark, L. O., Farthing, C. and Sims, A., ‘Simplicity of algebras associated to étale groupoids’, Semigroup Forum 88(2) (2014), 433452.
[12] Brown, J. H., Clark, L. O. and an Huef, A., ‘Purely infinite simple Steinberg algebras have purely infinite simple $C^{\ast }$ -algebras’, Preprint, 2017, arXiv e-prints.
[13] Brown, J. H. and an Huef, A., ‘The socle and semisimplicity of a Kumjian–Pask algebra’, Comm. Algebra 43(7) (2015), 27032723.
[14] Carlsen, T. M. and Rout, J., ‘Diagonal-preserving graded isomorphisms of Steinberg algebras’, Commun. Contemp. Math., to appear.
[15] Carlsen, T. M., Ruiz, E. and Sims, A., ‘Equivalence and stable isomorphism of groupoids, and diagonal-preserving stable isomorphisms of graph C -algebras and Leavitt path algebras’, Proc. Amer. Math. Soc. 145(4) (2017), 15811592.
[16] Clark, L. O., Barquero, D. M., González, C. M. and Siles Molina, M., ‘Using the Steinberg algebra model to determine the center of any Leavitt path algebra’, Preprint, 2016, arXiv e-prints.
[17] Clark, L. O., Barquero, D. M., Gonzalez, C. M. and Siles Molina, M., ‘Using Steinberg algebras to study decomposability of Leavitt path algebras’, Forum Math. 29(6) (2017), 13111324.
[18] Clark, L. O. and Edie-Michell, C., ‘Uniqueness theorems for Steinberg algebras’, Algebr. Represent. Theory 18(4) (2015), 907916.
[19] Clark, L. O., Edie-Michell, C., an Huef, A. and Sims, A., ‘Ideals of Steinberg algebras of strongly effective groupoids, with applications to Leavitt path algebras’, Preprint, 2016, arXiv e-prints.
[20] Clark, L. O., Exel, R. and Pardo, E., ‘A generalised uniqueness theorem and the graded ideal structure of Steinberg algebras’, Preprint, 2016, arXiv e-prints.
[21] Clark, L. O., Farthing, C., Sims, A. and Tomforde, M., ‘A groupoid generalisation of Leavitt path algebras’, Semigroup Forum 89(3) (2014), 501517.
[22] Clark, L. O. and Sims, A., ‘Equivalent groupoids have Morita equivalent Steinberg algebras’, J. Pure Appl. Algebra 219(6) (2015), 20622075.
[23] Connell, I. G., ‘On the group ring’, Canad. J. Math. 15 (1963), 650685.
[24] Exel, R., ‘Inverse semigroups and combinatorial C -algebras’, Bull. Braz. Math. Soc. (N.S.) 39(2) (2008), 191313.
[25] Hazrat, R. and Li, H., ‘Graded Steinberg algebras and partial actions’, Preprint, 2017, arXiv e-prints.
[26] Lawson, M. V., Inverse Semigroups: The Theory of Partial Symmetries (World Scientific, River Edge, NJ, 1998).
[27] Mitchell, B., ‘Rings with several objects’, Adv. Math. 8 (1972), 1161.
[28] Nystedt, P., Öinert, J. and Pinedo, H., ‘Artinian and noetherian partial skew groupoid rings’, Preprint, 2016, arXiv e-prints.
[29] Okniński, J., ‘Noetherian property for semigroup rings’, in: Ring Theory (Granada, 1986), Lecture Notes in Mathematics, 1328 (Springer, Berlin, 1988), 209218.
[30] Okniński, J., Semigroup Algebras, Monographs and Textbooks in Pure and Applied Mathematics, 138 (Marcel Dekker, New York, 1991).
[31] Paterson, A. L. T., Groupoids, Inverse Semigroups, and Their Operator Algebras, Progress in Mathematics, 170 (Birkhäuser, Boston, MA, 1999).
[32] Renault, J., A Groupoid Approach to C -Algebras, Lecture Notes in Mathematics, 793 (Springer, Berlin, 1980).
[33] Resende, P., ‘Étale groupoids and their quantales’, Adv. Math. 208(1) (2007), 147209.
[34] Steinberg, B., ‘Möbius functions and semigroup representation theory. II. Character formulas and multiplicities’, Adv. Math. 217(4) (2008), 15211557.
[35] Steinberg, B., ‘A groupoid approach to discrete inverse semigroup algebras’, Preprint, 2009, arXiv:0903.3456.
[36] Steinberg, B., ‘A groupoid approach to discrete inverse semigroup algebras’, Adv. Math. 223(2) (2010), 689727.
[37] Steinberg, B., ‘Modules over étale groupoid algebras as sheaves’, J. Aust. Math. Soc. 97(3) (2014), 418429.
[38] Steinberg, B., Representation Theory of Finite Monoids, Universitext (Springer, Cham, 2016).
[39] Steinberg, B., ‘Simplicity, primitivity and semiprimitivity of étale groupoid algebras with applications to inverse semigroup algebras’, J. Pure Appl. Algebra 220(3) (2016), 10351054.
[40] Steinberg, B., ‘Diagonal-preserving isomorphisms of étale groupoid algebras’, Preprint, 2017, arXiv e-prints.
[41] Webb, P., ‘An introduction to the representations and cohomology of categories’, in: Group Representation Theory (EPFL Press, Lausanne, 2007), 149173.
[42] Zelmanov, E. I., ‘Semigroup algebras with identities’, Sibirsk. Mat. Zh. 18(4) (1977), 787798, 956.
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