{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T02:46:13Z","timestamp":1767926773731,"version":"3.49.0"},"reference-count":28,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2013,10,8]],"date-time":"2013-10-08T00:00:00Z","timestamp":1381190400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Struct. Comp. Sci."],"published-print":{"date-parts":[[2014,4]]},"abstract":"<jats:p>We give a constructive proof showing that every finitely generated polynomial ideal has a Gr\u00f6bner basis, provided the ring of coefficients is Noetherian in the sense of Richman and Seidenberg. That is, we give a constructive termination proof for a variant of the well-known algorithm for computing the Gr\u00f6bner basis. In combination with a purely order-theoretic result we have proved in a separate paper, this yields a unified constructive proof of the Hilbert basis theorem for all Noether classes: if a ring belongs to a Noether class, then so does the polynomial ring. Our proof can be seen as a constructive reworking of one of the classical proofs, in the spirit of the partial realisation of Hilbert's programme in algebra put forward by Coquand and Lombardi. The rings under consideration need not be commutative, but are assumed to be coherent and strongly discrete: that is, they admit a membership test for every finitely generated ideal. As a complement to the proof, we provide a prime decomposition for commutative rings possessing the finite-depth property.<\/jats:p>","DOI":"10.1017\/s0960129513000509","type":"journal-article","created":{"date-parts":[[2013,10,8]],"date-time":"2013-10-08T12:55:58Z","timestamp":1381236958000},"source":"Crossref","is-referenced-by-count":8,"title":["Constructing Gr\u00f6bner bases for Noetherian rings"],"prefix":"10.1017","volume":"24","author":[{"given":"HERV\u00c9","family":"PERDRY","sequence":"first","affiliation":[]},{"given":"PETER","family":"SCHUSTER","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2013,10,8]]},"reference":[{"key":"S0960129513000509_ref28","unstructured":"Zariski O. and Samuel P. (1958) Commutative Algebra I, Van Nostrand."},{"key":"S0960129513000509_ref26","unstructured":"Tennenbaum J. (1973) A Constructive Version of Hilbert's Basis Theorem, Ph.D. thesis, University of California San Diego."},{"key":"S0960129513000509_ref25","doi-asserted-by":"publisher","DOI":"10.1007\/BF02925651"},{"key":"S0960129513000509_ref22","doi-asserted-by":"publisher","DOI":"10.1081\/AGB-120018518"},{"key":"S0960129513000509_ref20","doi-asserted-by":"publisher","DOI":"10.1016\/0020-0190(88)90126-3"},{"key":"S0960129513000509_ref16","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4419-8640-5"},{"key":"S0960129513000509_ref15","doi-asserted-by":"publisher","DOI":"10.1007\/s00209-011-0847-1"},{"key":"S0960129513000509_ref9","doi-asserted-by":"publisher","DOI":"10.1016\/j.jalgebra.2010.04.014"},{"key":"S0960129513000509_ref5","doi-asserted-by":"publisher","DOI":"10.1017\/S0960129506005627"},{"key":"S0960129513000509_ref18","doi-asserted-by":"publisher","DOI":"10.1002\/malq.200710042"},{"key":"S0960129513000509_ref6","first-page":"33","article-title":"Gr\u00f6bner bases in type theory","volume":"1657","author":"Coquand","year":"1999","journal-title":"Types for proofs and programs, Proceedings, TYPES, Irsee 1998"},{"key":"S0960129513000509_ref21","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1974-0416874-9"},{"key":"S0960129513000509_ref11","doi-asserted-by":"publisher","DOI":"10.1016\/S0747-7171(08)80154-X"},{"key":"S0960129513000509_ref24","doi-asserted-by":"publisher","DOI":"10.1007\/11780342_49"},{"key":"S0960129513000509_ref23","doi-asserted-by":"publisher","DOI":"10.1109\/LICS.2012.68"},{"key":"S0960129513000509_ref27","doi-asserted-by":"publisher","DOI":"10.1016\/j.jalgebra.2006.01.051"},{"key":"S0960129513000509_ref4","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0021089"},{"key":"S0960129513000509_ref17","doi-asserted-by":"publisher","DOI":"10.1016\/j.jsc.2003.02.001"},{"key":"S0960129513000509_ref13","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511565847.023"},{"key":"S0960129513000509_ref2","first-page":"326","volume-title":"Proceedings of the Ninetenth Annual IEEE Symposium on Logic in Computer Science","author":"Berger","year":"2004"},{"key":"S0960129513000509_ref19","doi-asserted-by":"publisher","DOI":"10.1017\/S0960129510000460"},{"key":"S0960129513000509_ref10","first-page":"294","article-title":"A direct proof of Wiener's theorem","volume":"7318","author":"Hendtlass","year":"2012","journal-title":"How the World Computes: Proceedings, CiE 2012, Turing Centenary Conference and Eighth Conference on Computability in Europe, Cambridge"},{"key":"S0960129513000509_ref1","doi-asserted-by":"publisher","DOI":"10.1090\/gsm\/003\/02"},{"key":"S0960129513000509_ref3","volume-title":"Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal","author":"Buchberger","year":"1965"},{"key":"S0960129513000509_ref7","volume-title":"Essays in Constructive Mathematics","author":"Edwards","year":"2005"},{"key":"S0960129513000509_ref8","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0084570"},{"key":"S0960129513000509_ref12","volume-title":"Commutative Rings","author":"Kaplansky","year":"1974"},{"key":"S0960129513000509_ref14","volume-title":"Modules projectifs de type fini","author":"Lombardi","year":"2011"}],"container-title":["Mathematical Structures in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0960129513000509","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,22]],"date-time":"2019-04-22T21:27:06Z","timestamp":1555968426000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0960129513000509\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,10,8]]},"references-count":28,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2014,4]]}},"alternative-id":["S0960129513000509"],"URL":"https:\/\/doi.org\/10.1017\/s0960129513000509","relation":{},"ISSN":["0960-1295","1469-8072"],"issn-type":[{"value":"0960-1295","type":"print"},{"value":"1469-8072","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,10,8]]},"article-number":"e240206"}}