On Proper Class Coprojectively Generated by Modules With Projective Socle

Year 2020, Issue: 32, 79 - 87, 30.09.2020

Ayşe Çobankaya

Abstract

Let $\varepsilon$ : 0 --> A -->^f B -->^g C --> 0 be a short exact sequence of modules and module homomorphism. $\varepsilon$ is called gd-closed sequence if Imf is gd-closed in B. In this paper, the proper class $GD$− Closed, which is coprojectively generated by modules with projective socle, be studied and also its relations among Neat, Closed, $D$−Closed, $S$−Closed be investigated. Additionally, we examine coprojective modules of this class.                                                                 

                                                                                                                                                                                                                                                                      .

Keywords

Gd-closed,, g-semartinian modules, G-Dickson torsion theory, gdc-flat modules

Supporting Institution

Research Fund of the Cukurova University.

Project Number

Project number: 12308

References

• N.V. Dung, D.V. Huynh, P.F. Smith, R. Wisbauer, Extending modules, Pitman Research Notes in Math. Ser. 313 Longman Scientific and Technical, Harlow, 1994.
• E. Büyükaşık, Y. Durgun, Neat-flat Modules, Comm. Algebra 44 (2016) 416-428.
• E. Büyükaşık, Y. Durgun, Absolutely s-pure modules and neat-flat modules, Comm. Algebra 43 (2)(2015) 384-399.
• A. David Buchsbaum, A note on homology in categories, Ann. of Math. (69) (2) (1959) 66-74.
• J. Clark, C. Lomp, N. Vanaja, and R. Wisbauer, Lifting modules, Birkh¨auser Verlag, Basel, 2006.
• S. Crivei, S.S¸ahinkaya, Modules whose closed submodules with essential socle are direct summands, Taiwanese J. Math. 18(4)(2014) 989-1002.
• Y. Durgun, A. Çobankaya, On subprojectivity domains of g-semiartinian modules, J. Algebra Appl. (2021) https://doi.org/10.1142/S021949882150119X, (in press).
• Y. Durgun, A. Çobankaya, Proper classes generated by t-closed submodules, An. S¸t. Univ. Ovidius Constanta 27 (2019) 83-95.
• Y. Durgun, A. Çobankaya, G-Dickson Torsion Theory, International Science, Mathematics and Engineering Sciences Congress (2019) 978-983.
• P. M. Cohn, On the free product of associative rings Math. Z. 71(1959) 380-398.
• Y. Durgun, S. Özdemir, On S-Closed Submodules, J. Korean Math. Soc. 54 (2017) 1281-1299.
• Y. Durgun, S. Özdemir, On D-Closed Submodules, Proc. Indian Acad. Sci (Math. Sci.) 130 (1)(2020) 14pp.
• Y. Durgun, D-Extending Modules, Hacet. J. Math. Stat. (49)(2020) 1-7, https://doi.org/10.15672/hujms. 460241.
• E. E. Enochs, O. M. G. Jenda, Relative Homological Algebra Walter de Gruyter, Berlin-New York, 2000.
• L. Fuchs, Neat submodules over integral domains, Period. Math. Hungar. 64(2)(2012) 131-143.
• K.R. Goodearl, Singular torsion and the splitting properties, Amer. Math. Soc. 124 , Providence, R. I. 1972.
• A. I. Generalov, On weak and !-high purities in the category of modules, Mat. Sb. (N.S.) 105(147)(3) (1978), 389-402.
• Y. Kara, A. Tercan, When some complement of a z-closed submodule is a summand, Comm. Algebra, 46(7)(2018) 3071-3078.
• Kepka, T., On one class of purities, Comment. Math. Univ. Carolinae, 14 (1973), 139-154.
• W.K. Nicholson, J.F. Watters, Rings with projective socle, Proc. Amer. Math. Soc., 102 (1988) 443-450.
• G. Renault, ´Etude de certains anneaux li´es aux sous-modules compl´ements d’un A-module, C. R. Acad. Sci. Paris 259 (1964) 4203-4205.
• J. Rotman, An Introduction to Homological Algebra, Universitext, Springer-Verlag, New York, 2009.
• E.G. Skljarenko, Relative homological algebra in the category of modules, Russian Math. Surveys, 333(201) (1978) 85-120.
• H. Z¨oschinger, Schwach-Flache Moduln. Comm. Algebra 41 (12) (2013) 4393-4407.