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Czechoslovak Mathematical Journal, Vol. 68, No. 3, pp. 677-686, 2018

Strict Mittag-Leffler conditions and locally split morphisms

Yanjiong Yang, Xiaoguang Yan

Received December 5, 2016.   First published March 6, 2018.
Abstract:  In this paper, we prove that any pure submodule of a strict Mittag-Leffler module is a locally split submodule. As applications, we discuss some relations between locally split monomorphisms and locally split epimorphisms and give a partial answer to the open problem whether Gorenstein projective modules are Ding projective.
Keywords:  strict Mittag-Leffler condition; locally split morphism; Gorenstein projective module; Ding projective module
Classification MSC:  13D02, 13D07, 13E05, 16D10, 16D80, 16D90
DOI:  10.21136/CMJ.2018.0621-16
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References:
[1] L. Angeleri Hügel, S. Bazzoni, D. Herbera: A solution to the Baer splitting problem. Trans. Am. Math. Soc. 360 (2008), 2409-2421. DOI 10.1090/S0002-9947-07-04255-9 | MR 2373319 | Zbl 1136.13005
[2] L. Angeleri Hügel, D. Herbera: Mittag-Leffler conditions on modules. Indiana Univ. Math. J. 57 (2008), 2459-2517. DOI 10.1512/iumj.2008.57.3325 | MR 2463975 | Zbl 1206.16002
[3] G. Azumaya: Locally pure-projective modules. Azumaya Algebras, Actions, and Modules Proc. Conf., Bloomington 1990, Contemp. Math. 124, Am. Math. Soc., Providence, 1992, 17-22. DOI 10.1090/conm/124 | MR 1144024 | Zbl 0743.16003
[4] S. Bazzoni, D. Herbera: One dimensional tilting modules are of finite type. Algebr. Represent. Theory 11 (2008), 43-61. DOI 10.1007/s10468-007-9064-3 | MR 2369100 | Zbl 1187.16008
[5] S. Bazzoni, J. Šťovíček: All tilting modules are of finite type. Proc. Am. Math. Soc. 135 (2007), 3771-3781. DOI 10.1090/S0002-9939-07-08911-3 | MR 2341926 | Zbl 1132.16008
[6] D. Bennis, N. Mahdou: Strongly Gorenstein projective, injective, and flat modules. J. Pure Appl. Algebra 210 (2007), 437-445. DOI 10.1016/j.jpaa.2006.10.010 | MR 2320007 | Zbl 1118.13014
[7] S. U. Chase: Direct products of modules. Trans. Am. Math. Soc. 97 (1960), 457-473. DOI 10.2307/1993382 | MR 0120260 | Zbl 0100.26602
[8] N. Ding, Y. Li, L. Mao: Strongly Gorenstein flat modules. J. Aust. Math. Soc. 86 (2009), 323-338. DOI 10.1017/S1446788708000761 | MR 2529328 | Zbl 1200.16010
[9] I. Emmanouil: On certain cohomological invariants of groups. Adv. Math. 225 (2010), 3446-3462. DOI 10.1016/j.aim.2010.06.007 | MR 2729012 | Zbl 1214.20050
[10] I. Emmanouil, O. Talelli: On the flat length of injective modules. J. Lond. Math. Soc., II. Ser. 84 (2011), 408-432. DOI 10.1112/jlms/jdr014 | MR 2835337 | Zbl 1237.16004
[11] E. E. Enochs, O. M. G. Jenda: Relative Homological Algebra. De Gruyter Expositions in Mathematics 30, Walter de Gruyter, Berlin (2000). DOI 10.1515/9783110803662 | MR 1753146 | Zbl 0952.13001
[12] J. Gillespie: Model structures on modules over Ding-Chen rings. Homology Homotopy Appl. 12 (2010), 61-73. DOI 10.4310/HHA.2010.v12.n1.a6 | MR 2607410 | Zbl 1231.16005
[13] A. Grothendieck: Techniques de construction et théorèmes d'existence en géométrie algébrique. III. Préschemas quotients. Sém. Bourbaki, Vol. 6, 1960-1961. Soc. Math. France, Paris, 1995, 99-118. (In French.) MR 1611786 | Zbl 0235.14007
[14] D. Herbera, J. Trlifaj: Almost free modules and Mittag-Leffler conditions. Adv. Math. 229 (2012), 3436-3467. DOI 10.1016/j.aim.2012.02.013 | MR 2900444 | Zbl 1279.16001
[15] M. Raynaud, L. Gruson: Critères de platitude et de projectivité. Techniques de "platification" d'un module. Invent. Math. 13 (1971), 1-89. (In French.) DOI 10.1007/BF01390094 | MR 0308104 | Zbl 0227.14010
[16] W. Zimmermann: On locally pure-injective modules. J. Pure Appl. Algebra 166 (2002), 337-357. DOI 10.1016/S0022-4049(01)00011-1 | MR 1870625 | Zbl 0993.16002
Affiliations:   Yanjiong Yang, Department of Mathematics, Taizhou University, 93 Jichuan East Rd, hailing, Taizhou 225300, Jiangsu, China, e-mail: yanjiong1019@126.com; Xiaoguang Yan, School of Information Technology, Nanjing Xiaozhuang University, 3601 Hongjing Avenue, Jiangning, Nanjing 211171, Jiangsu, China, e-mail: yanxg1109@sina.cn
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