{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T16:43:10Z","timestamp":1775839390622,"version":"3.50.1"},"reference-count":27,"publisher":"Springer Science and Business Media LLC","issue":"5-6","license":[{"start":{"date-parts":[[2006,11,18]],"date-time":"2006-11-18T00:00:00Z","timestamp":1163808000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Appl Categor Struct"],"published-print":{"date-parts":[[2006,11,29]]},"DOI":"10.1007\/s10485-006-9043-6","type":"journal-article","created":{"date-parts":[[2006,11,17]],"date-time":"2006-11-17T12:50:52Z","timestamp":1163767852000},"page":"431-469","source":"Crossref","is-referenced-by-count":27,"title":["Cleft Extensions of Hopf Algebroids"],"prefix":"10.1007","volume":"14","author":[{"given":"Gabriella","family":"B\u00f6hm","sequence":"first","affiliation":[]},{"given":"Tomasz","family":"Brzezi\u0144ski","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2006,11,18]]},"reference":[{"key":"9043_CR1","first-page":"1","volume-title":"Hopf Algebras in Noncommutative Geometry and Physics. Lecture Notes in Pure and Appl. Math., vol.\u00a0239","author":"J. Abuhlail","year":"2005","unstructured":"Abuhlail, J.: Morita contexts for corings and equivalences. In: Caenepeel, S., Van Oystaeyen, F. (eds.) Hopf Algebras in Noncommutative Geometry and Physics. Lecture Notes in Pure and Appl. Math., vol.\u00a0239, pp. 1\u201319. Marcel Dekker, New York (2005)"},{"key":"9043_CR2","doi-asserted-by":"crossref","first-page":"679","DOI":"10.1016\/j.jalgebra.2004.07.043","volume":"284","author":"J.N. Alonso \u00c1lvarez","year":"2005","unstructured":"Alonso \u00c1lvarez, J.N., Fern\u00e1ndez Vilaboa, J.M., Gonz\u00e1lez Rodr\u00edguez, R., Rodr\u00edguez Raposo, A.B.: Weak C-cleft extensions, weak entwining structures and weak Hopf algebras. J. Algebra 284, 679\u2013704 (2005)","journal-title":"J. Algebra"},{"key":"9043_CR3","doi-asserted-by":"crossref","first-page":"276","DOI":"10.1016\/j.jalgebra.2005.09.012","volume":"299","author":"J.N. Alonso \u00c1lvarez","year":"2006","unstructured":"Alonso \u00c1lvarez, J.N., Fern\u00e1ndez Vilaboa, J.M., Gonz\u00e1lez Rodr\u00edguez, R., Rodr\u00edguez Raposo, A.B.: Weak C-cleft extensions and weak Galois extensions. J. Algebra 299, 276\u2013293 (2006)","journal-title":"J. Algebra"},{"key":"9043_CR4","doi-asserted-by":"crossref","first-page":"520","DOI":"10.1016\/j.jalgebra.2005.09.023","volume":"296","author":"I. B\u00e1lint","year":"2006","unstructured":"B\u00e1lint, I., Szlach\u00e1nyi, K.: Finitary Galois extensions over non-commutative bases. J. Algebra 296, 520\u2013560 (2006)","journal-title":"J. Algebra"},{"key":"9043_CR5","doi-asserted-by":"crossref","first-page":"671","DOI":"10.1090\/S0002-9947-1986-0860387-X","volume":"298","author":"R.J. Blattner","year":"1986","unstructured":"Blattner, R.J., Cohen, M., Montgomery, S.: Crossed products and inner actions of Hopf algebras. Trans. Amer. Math. Soc. 298, 671\u2013711 (1986)","journal-title":"Trans. Amer. Math. Soc."},{"key":"9043_CR6","doi-asserted-by":"crossref","first-page":"37","DOI":"10.2140\/pjm.1989.137.37","volume":"137","author":"R.J. Blattner","year":"1989","unstructured":"Blattner, R.J., Montgomery, S.: Crossed products and Galois extensions of Hopf algebras. Pacific J. Math. 137, 37\u201354 (1989)","journal-title":"Pacific J. Math."},{"issue":"4","key":"9043_CR7","doi-asserted-by":"crossref","first-page":"563","DOI":"10.1007\/s10468-005-8760-0","volume":"8","author":"G. B\u00f6hm","year":"2005","unstructured":"B\u00f6hm, G.: Integral theory for Hopf algebroids. Alg. Rep. Theory 8(4), 563\u2013599 (2005)","journal-title":"Alg. Rep. Theory"},{"key":"9043_CR8","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1007\/BF02824833","volume":"51","author":"G. B\u00f6hm","year":"2005","unstructured":"B\u00f6hm, G.: Galois theory for Hopf algebroids. Ann. Univ. Ferrara, Sez. 7: Sci. Mat. 51, 233\u2013262 (2005)","journal-title":"Ann. Univ. Ferrara, Sez. 7: Sci. Mat."},{"key":"9043_CR9","doi-asserted-by":"crossref","first-page":"2579","DOI":"10.1155\/IMRN.2005.2579","volume":"42","author":"G. B\u00f6hm","year":"2005","unstructured":"B\u00f6hm, G., Brzezi\u0144ski, T.: Strong connections and the relative Chern\u2013Galois character for corings. Internat. Math. Res. Notices 42, 2579\u20132625 (2005)","journal-title":"Internat. Math. Res. Notices"},{"key":"9043_CR10","doi-asserted-by":"crossref","first-page":"708","DOI":"10.1016\/j.jalgebra.2003.09.005","volume":"274","author":"G. B\u00f6hm","year":"2004","unstructured":"B\u00f6hm, G., Szlach\u00e1nyi, K.: Hopf algebroids with bijective antipodes: axioms, integrals and duals. J. Algebra 274, 708\u2013750 (2004)","journal-title":"J. Algebra"},{"key":"9043_CR11","doi-asserted-by":"crossref","unstructured":"B\u00f6hm, G., Vercruysse, J.: Morita theory for coring extensions and cleft bicomodules. arxiv:math.RA\/0601464 DOI: 10.1016\/J.AIM.2006.05.010","DOI":"10.1016\/j.aim.2006.05.010"},{"key":"9043_CR12","doi-asserted-by":"crossref","first-page":"290","DOI":"10.1006\/jabr.1998.7738","volume":"215","author":"T. Brzezi\u0144ski","year":"1999","unstructured":"Brzezi\u0144ski, T.: On modules associated to coalgebra Galois extensions. J. Algebra 215, 290\u2013317 (1999)","journal-title":"J. Algebra"},{"key":"9043_CR13","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1007\/BF02824821","volume":"51","author":"T. Brzezi\u0144ski","year":"2005","unstructured":"Brzezi\u0144ski, T.: A note on coring extensions. Ann. Univ. Ferrara, Sez. 7: Sci. Mat. 51, 15\u201327 (2005)","journal-title":"Ann. Univ. Ferrara, Sez. 7: Sci. Mat."},{"key":"9043_CR14","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1016\/S0022-4049(02)00128-7","volume":"175","author":"T. Brzezi\u0144ski","year":"2002","unstructured":"Brzezi\u0144ski, T., Caenepeel, S., Militaru, G.: Doi-Koppinen modules for quantum groupoids. J. Pure Appl. Algebra 175, 45\u201362 (2002)","journal-title":"J. Pure Appl. Algebra"},{"key":"9043_CR15","doi-asserted-by":"crossref","first-page":"1347","DOI":"10.1080\/00927879908826498","volume":"27","author":"T. Brzezi\u0144ski","year":"1999","unstructured":"Brzezi\u0144ski, T., Hajac, P.M.: Coalgebra extensions and algebra coextensions of Galois type. Comm. Algebra 27, 1347\u20131367 (1999)","journal-title":"Comm. Algebra"},{"key":"9043_CR16","doi-asserted-by":"crossref","first-page":"467","DOI":"10.1007\/s002200050274","volume":"191","author":"T. Brzezi\u0144ski","year":"1998","unstructured":"Brzezi\u0144ski, T., Majid, S.: Coalgebra bundles. Comm. Math. Phys. 191, 467\u2013492 (1998)","journal-title":"Comm. Math. Phys."},{"key":"9043_CR17","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511546495","volume-title":"Corings and Comodules","author":"T. Brzezi\u0144ski","year":"2003","unstructured":"Brzezi\u0144ski, T., Wisbauer, R.: Corings and Comodules. Cambridge University Press, Cambridge, UK (2003)"},{"key":"9043_CR18","doi-asserted-by":"crossref","unstructured":"Caenepeel, S., De Groot, E.: Modules over weak entwining structures. In: Andruskiewitsch, N., Ferrer Santos, W.R., Schneider, H.-J. (eds) New Trends in Hopf Algebra Theory. Contemp. Math.- Amer. Math. Soc., vol 267, pp. 31\u201354. Providence, RI (2000)","DOI":"10.1090\/conm\/267\/04263"},{"key":"9043_CR19","doi-asserted-by":"crossref","first-page":"210","DOI":"10.1016\/j.jalgebra.2004.02.002","volume":"276","author":"S. Caenepeel","year":"2004","unstructured":"Caenepeel, S., Vercruysse, J., Wang, S.: Morita theory for corings and cleft entwining structures. J. Algebra 276, 210\u2013235 (2004)","journal-title":"J. Algebra"},{"key":"9043_CR20","first-page":"801","volume":"14","author":"Y. Doi","year":"1986","unstructured":"Doi, Y., Takeuchi, M.: Cleft comodule algebras for a bialgebra. Comm. Algebra 14, 801\u2013817 (1986)","journal-title":"Comm. Algebra"},{"key":"9043_CR21","doi-asserted-by":"crossref","unstructured":"Kadison, L.: Depth two and the Galois coring. In: Fuchs, J., Stolin, A.A. (eds.) Noncommutative geometry and representation theory in mathematical physics. Contemp. Math.-Amer. Math. Soc., vol.\u00a0391 pp. 149\u2013156. Providence, RI (2005)","DOI":"10.1090\/conm\/391\/07325"},{"key":"9043_CR22","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1016\/j.jalgebra.2006.01.013","volume":"305","author":"L. Kadison","year":"2006","unstructured":"Kadison, L.: An endomorphism ring theorem for Galois and D2 extensions. J. Algebra 305, 163\u2013184 (2006)","journal-title":"J. Algebra"},{"key":"9043_CR23","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1016\/S0001-8708(02)00028-2","volume":"179","author":"L. Kadison","year":"2003","unstructured":"Kadison, L., Szlach\u00e1nyi, K.: Bialgebroid actions on depth two extensions and duality. Adv. Math. 179, 75\u2013121 (2003)","journal-title":"Adv. Math."},{"key":"9043_CR24","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1142\/S0129167X96000050","volume":"7","author":"J.-H. Lu","year":"1996","unstructured":"Lu, J.-H.: Hopf algebroids and quantum groupoids. Internat. J. Math. 7, 47\u201370 (1996)","journal-title":"Internat. J. Math."},{"key":"9043_CR25","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1023\/A:1008608028634","volume":"6","author":"P. Schauenburg","year":"1998","unstructured":"Schauenburg, P.: Bialgebras over noncommutative rings and a structure theorem for Hopf bimodules. Appl. Categ. Structures 6, 193\u2013222 (1998)","journal-title":"Appl. Categ. Structures"},{"key":"9043_CR26","volume-title":"Hopf Algebras","author":"M.E. Sweedler","year":"1969","unstructured":"Sweedler, M.E.: Hopf Algebras. Benjamin, New York (1969)"},{"key":"9043_CR27","doi-asserted-by":"crossref","first-page":"459","DOI":"10.2969\/jmsj\/02930459","volume":"29","author":"M. Takeuchi","year":"1977","unstructured":"Takeuchi, M.: Groups of algebras over A\u2297\u0100. J. Math. Soc. Japan 29, 459\u2013492 (1977)","journal-title":"J. Math. Soc. Japan"}],"container-title":["Applied Categorical Structures"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10485-006-9043-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10485-006-9043-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10485-006-9043-6","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,8]],"date-time":"2024-02-08T09:43:49Z","timestamp":1707385429000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10485-006-9043-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,11,18]]},"references-count":27,"journal-issue":{"issue":"5-6","published-print":{"date-parts":[[2006,11,29]]}},"alternative-id":["9043"],"URL":"https:\/\/doi.org\/10.1007\/s10485-006-9043-6","relation":{},"ISSN":["0927-2852","1572-9095"],"issn-type":[{"value":"0927-2852","type":"print"},{"value":"1572-9095","type":"electronic"}],"subject":[],"published":{"date-parts":[[2006,11,18]]}}}