Abstract
Inspired by the notion of virtually extending module, in this work we have proposed and investigated the behaviours of virtually lifting module as the dual of virtually extending module. Several properties of this structure have been discussed and got that virtually lifting module is a \(D_{12}\) module. It is observed that for a V ring R, a module M is virtually lifting iff it is lifting. Moreover, we have defined virtually \(D_{2}\) module as a proper extension of \(D_{2}\) module, and seen that virtually \(D_{2}\) modules is inherited by direct summands. Finally, we have investigated principally h-lifting module and proved that finite direct sum of principally h-lifting modules are principally h-lifting if each components are principally relatively projective.
Similar content being viewed by others
References
Acar, U., Harmanci, A.: Principally supplemented modules. Albanian J. Math. 4, 79–88 (2010)
Akalan, E., Birkenmeier, G.F., Tercan, A.: Goldie extending modules. Commun. Algebra 37, 663–683 (2009)
Atani, S.E., Khoramdel, M., Hesari, S.: On strongly extending modules. Kyungpook Math. J. 54, 237–247 (2014)
Behboodi, M., Daneshvar, A., Vedadi, M.R.: Several generalizations of the Wedderburn–Artin theorem with applications. Algebras Represent. Theory 21, 1333–1342 (2018)
Behboodi, M., Daneshvar, A., Vedadi, M.R.: Virtually semisimple modules and a generalization of the Wedderburn–Artin theorem. Commun. Algebra 46, 2384–2395 (2018)
Behboodi, M., Daneshvar, A., Vedadi, M.R.: Structure of virtually semisimple modules over commutative rings. Commun. Algebra 48, 2872–2882 (2020)
Behboodi, M., Moradzadeh-Dehkordi, A., Nejadi, M.Q.: Virtually uniserial modules and rings. J. Algebra 549, 365–385 (2020)
Behboodi, M., Moradzadeh-Dehkordi, A., Qourchi Nejadi, M.: Virtually homo-uniserial modules and rings. Commun. Algebra 49, 3837–3849 (2021)
Clark, J., Wisbauer, R.: $\Sigma $-extending modules. J. Pure Appl. Algebra 104, 19–32 (1995)
Clark, J., Lomp, C., Vanaja, N., Wisbauer, R.: Lifting Modules, Supplements and Projectivity in Module Theory. Frontiers in Mathematics, Birkhauser Verlag, Basel (2006)
Colby, R.R., Rutter, E.A.: Generalizations of QF-3 algebras. Trans. Am. Math. Soc. 153, 371–386 (1971)
Dung, N.V., Van Huynh, D., Smith, P.F., Wisbauer, R.: Extending Modules. Routledge, London (1994)
Ganesan, L., Vanaja, N.: Modules for which every submodule has a unique coclosure. Commun. Algebra 30, 2355–2377 (2002)
Gang, Y., Zhong-Kui, L.: On Hopfian and co-Hopfian modules. Vietnam J. Math. 35, 73–80 (2007)
Ghorbani, A., Haghany, A.: Generalized Hopfian modules. J. Algebra 255, 324–341 (2002)
Idelhadj, A., Tribak, R.: On some properties of $\oplus $-supplemented modules. Int. J. Math. Math. Sci. 2003, 4373–4387 (2003)
Kamal, M.A., Elmnophy, O.A.: On $P$-extending modules. Acta Math. Univ. Comen. N. Ser. 74, 279–286 (2005)
Kamal, M.A., Yousef, A.: On principally lifting modules. Int. Electron. J. Algebra 2, 127–137 (2007)
Karabacak, F., Kosan, M.T., Quynh, T.C., Tasdemir, O.: On modules and rings in which complements are isomorphic to direct summands. Commun. Algebra 50, 1154–1168 (2022)
Keskin, D.: On lifting modules. Commun. Algebra 28, 3427–3440 (2000)
Mohamed, S.H., Müller, B.J.: Continuous and Discrete Modules. Cambridge University Press, Cambridge (1990)
Orhan, N., Tütüncü, D.K., Tribak, R.: On hollow-lifting modules. Taiwan. J. Math. 11, 545–568 (2007)
Oshiro, K.: Lifting modules, extending modules and their applications to generalized uniserial rings. Hokkaido Math. J. 13, 339–346 (1984)
Patel, M.K., Choubey, S.K., Das, L.K.: On finitely-hollow-weak lifting modules. Proc. Indian Natl Sci. Acad. 87, 143–147 (2021)
Tütüncü, D.K., Tribak, R.: On D12-modules. Rocky Mt. J. Math. 43(4), 1355–1373 (2013)
Ungor, B., Halicioglu, S., Harmanci, A.: On a class of $\oplus $-supplemented modules Ring Theory and Its Applications. Contemp. Math. 609, 123–136 (2014)
Wisbauer, R.: Foundations of Module and Ring Theory. Gordon and Breach, Reading (1991)
Xue, W.: Characterization of rings using direct-projective modules and direct-injective modules. J. Pure Appl. Algebra 87(1), 99–104 (1993)
Acknowledgements
M. K. Patel would like to thank NBHM, D.A.E., for financial support with File No. 02211/3/2019 NBHM (R.P.) RD-II/1439. The authors are grateful to the referee for their precious suggestions, which has helped us to improve the quality of the paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Das, L.K., Patel, M.K. On modules whose coclosed submodules are isomorphic to direct summands. Proc.Indian Natl. Sci. Acad. 88, 790–795 (2022). https://doi.org/10.1007/s43538-022-00131-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s43538-022-00131-z
Keywords
- Virtually semi-simple module
- \(VD_{1}\) module
- \(VC_{1}\) module
- \(VD_{2}\) module
- Principally h-lifting module