We introduce the notion of I -lifting modules as a proper generalization of the notion of lifting modules and present some properties of this class of modules. It is shown that if M is an I -lifting direct projective module, then S/▽ is regular and ▽ = JacS, where S is the ring of all R-endomorphisms of M and ▽ = {ϕ ∈ S | Im ϕ ≪ M}. Moreover, we prove that if M is a projective I -lifting module, then M is a direct sum of cyclic modules. The connections between I -lifting modules and dual Rickart modules are presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
E. P. Armendariz, “A note on extensions of Baer and P. P.-rings,” J. Austral. Math. Soc., 18, 470–473 (1974).
J. Clark, C. Lomp, N. Vanaja, and R. Wisbauer, “Lifting modules-supplements and projectivity in module theory,” Front. Math., Birkhäuser (2006).
S. Endo, “Note on p.p. rings,” Nagoya Math. J., 17, 167–170 (1960).
M. W. Evans, “On commutative P. P. rings,” Pacif. J. Math., 41, No. 3, 687–697 (1972).
I. Kaplansky, “Rings of operators,” Math. Lect. Note Ser., Benjamin, New York (1968).
D. Keskin, “On lifting modules,” Comm. Algebra, 28, No. 7, 3427–3440 (2000).
D. Keskin and W. Xue, “Generalizations of lifting modules,” Acta. Math. Hung., 91, 253–261 (2001).
D. Keskin and N. Orhan, “CCSR-modules and weak lifting modules,” East-West J. Math., 5, No. 1, 89–96 (2003).
M. T. Kosan, “δ-Lifting and δ-supplemented modules,” Algebra Colloq., 14, No. 1, 53–60 (2007).
G. Lee, S. T. Rizvi, and C. S. Roman, “Rickart modules,” Comm. Algebra, 38, No. 11, 4005–4027 (2010).
G. Lee, S. T. Rizvi, and C. S. Roman, “Dual Rickart modules,” Comm. Algebra, 39, 4036–4058 (2011).
Q. Liu and B. Y. Ouyang, “Rickart modules,” Nanjing Daxue Xuebao Shuxue Bannian Kan, 23, No. 1, 157–166 (2006).
Q. Liu, B. Y. Ouyang, and T. S. Wu, “Principally quasi-Baer modules,” J. Math. Res. Exposition, 29, No. 5, 823–830 (2009).
S. Maeda, “On a ring whose principal right ideals generated by idempotents form a lattice,” J. Sci. Hiroshima Univ. Ser. A, 24, 509–525 (1960).
S. H. Mohamed, and B. J. Müller, “Continuous and discrete modules,” London Math. Soc. Lect. Not. Ser. 147, Cambridge Univ. Press, Cambridge (1990).
S. T. Rizvi and C. S. Roman, “Baer property of modules and applications,” Adv. Ring Theory, 225–241 (2005).
Y. Talebi and T. Amouzegar, “Projective modules and a generalization of lifting modules,” East-West J. Math., 12, No. 1, 9–15 (2010).
D. K. Tütüncü and R. Tribak, “On T -noncosingular modules,” Bull. Austral. Math. Soc., 80, 462–471 (2009).
R. Wisbauer, Foundations of Module and Ring Theory, Gordon & Breach, Reading (1991).
Author information
Authors and Affiliations
Additional information
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 66, No. 11, pp. 1477–1484, November, 2014.
Rights and permissions
About this article
Cite this article
Amouzegar Kalati, T. A Generalization of Lifting Modules. Ukr Math J 66, 1654–1664 (2015). https://doi.org/10.1007/s11253-015-1042-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-015-1042-z