Article ID: MTJPAM-D-22-00037

Title: Goldie ss-supplemented modules

Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-22-00037; Volume 5 / Issue 1 / Year 2023, Pages 65-70

Document Type: Research Paper

Author(s): Fatih Gömleksiz a , Burcu Nişancı Türkmen b

aGraduate School of Natural and Applied Sciences, Department of Mathematics, Amasya University, Amasya, Turkey

bFaculty of Art and Science, Department of Mathematics, Ipekköy, Amasya University, Amasya, Turkey

Received: 14 November 2022, Accepted: 10 May 2023, Published: 11 June 2023

Corresponding Author: Fatih Gömleksiz (Email address:

Full Text: PDF


In this study, it has been determined the notion of Goldie ss-supplemented modules by the help of the relation \beta_{{ss}}^{*}, which is defined in the form W\beta_{{ss}}^{*}Z, which provides conditions both of \frac{W + Z}{W} \subseteq \frac{\text{Soc}_{s}(A) + W}{W} and \frac{W + Z}{Z} \subseteq \frac{\text{Soc}_{s}(A) + Z}{Z} for submodules W and Z of module A is an equivalence relation. The main features of Goldie ss-supplemented modules provided by this relation is examined. It is shown that the epimorphism \gamma:A \longrightarrow B provided the relation \beta_{{ss}}^{*} under certain conditions and the relation \beta_{{ss}}^{*} , is expressed in the maximal submodules. In addition, we obtain notions of Goldie ss-lifting modules using the relation \beta_{{ss}}^{*} and we prove several properties of notions of these modules.

Keywords: Socle of a module, semisimple module, ss-supplement, relation \beta_{{ss}}^{*}

  1. G. F. Birkenmeier, F. T. Mutlu, C. Nebiyev, N. Sökmez and A. Tercan, Goldie*-supplemented modules, Glasg. Math. J. 52 (A), 41–52, 2010.
  2. J. Clark, C. Lomp, N. Vanaja and R. Wisbauer, Lifting modules: Supplements and projectivity in module theory, Basel, Birkhauser, 2006.
  3. F. Eryilmaz, Ss-lifting modules and rings, Miskolc Math. Notes 22 (2), 655–662, 2020.
  4. F. Kasch, Modules and rings, Published for the London Mathematical Society by Academic Press Inc. (London) Ltd., 372, 1982.
  5. E. Kaynar, H. Çalişici and E. Türkmen, Ss-supplemented modules, Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat. 69 (1), 473–485, 2020.
  6. M. T. Koşan and D. Keskin, H-supplemented Duo modules, J. Algebra Appl. 6 (6), 965–971, 2007.
  7. A. Olgun and E. Türkmen, On a class of perfect rings, Honam Math. J. 42 (3), 591–600, 2020.
  8. D. W. Sharpe and P. Vamos, Injective modules, Lectures in Pure Mathematics University of Sheffield, The Great Britain, 190, 1972.
  9. Y. Talebi, A. R. M. Hamzekolaee and A.Tercan, Goldie-Rad-supplemented modules, An. Ştiinţ . Univ. “Ovidius" Constanţa Ser. Mat. 22 (3), 205–218, 2014.
  10. R. Wisbauer, Foundations of module and ring theory, Gordon and Breach, Philadelphia, 600, 1991.
  11. D. X. Zhou and X. R. Zhang, Small-essential submodules and morita duality, Southeast Asian Bull. Math. 35 (6), 1051–1062, 2011.