Erzincan University Journal of Science and Technology 2149-4584 Erzincan Binali Yıldırım Üniversitesi 10.18185/erzifbed.1251658 Engineering Mühendislik On tau-discrete modules https://orcid.org/0000-0001-7900-0529 Nişancı Türkmen Burcu AMASYA ÜNİVERSİTESİ https://orcid.org/0000-0002-7082-1176 Türkmen Ergül AMASYA UNIVERSITY 03 28 2024 17 1 37 43 02 15 2023 01 12 2024 Copyright © 2008, Erzincan Üniversitesi Fen Bilimleri Enstitüsü Dergisi 2008 Erzincan Üniversitesi Fen Bilimleri Enstitüsü Dergisi

An R-module M is said to be (quasi) τ-discrete if M is τ-lifting and has the property (D_2) (respectively, has the property (D_3)), where τ is a preradical in R-mod. It is shown that: (1) direct summands of a (quasi) τ-discrete module are (quasi) τ-discrete; (2) a projective module M is τ-discrete iff M/(τ(M)) is semisimple and τ(M) is QSL; (3) if a projective module M is Soc-lifting, then M/(Soc(M)) is Soc-discrete and Rad(M/Soc(M) ) is semisimple.

 preradical τ-lifting mdule (quasi) τ-discrete module. preradical τ-lifting mdule (quasi) τ-discrete module. This work was presented in 3rd International Conference on Engineering and Applied Natural Sciences (ICEANS 2023) on 14-17 January in 2023 at Konya/Turkey. [1] Al-Khazzi, I., and Smith, P.F., (1991) Modules with chain conditions on superfluous submodules, Communications in Algebra, Vol. 19(8), pp. 2331-2351. [2] Al-Takhman, K., Lomp, C. and Wisbauer, R. (2006) τ-complement and τ-supplement submodules, Algebra and Discrete Mathematics, 3, 1-15. [3] Alkan, M. (2009) On τ-lifting modules and τ-semiperfect modules, Turkish Journal of Mathematics, 33(2), 117-130. [4] Büyükaşık, E., Mermut, E. and Özdemir, S. (2010) Rad-supplemented modules, Rendiconti del Seminario Matematico della Universita di Padova, 124, 157-177. [5] Mohamed S.H. and Müller, B.J. (1990) Continuous and Discrete Modules”, London Mathematical Society, LNS 147 Cambridge University Press, Cambridge. 6] Nicholson, W.K. and Zhou, Y. (2005) Strongly lifting, Journal of Algebra, 285, 795-818. [7] Nişancı Türkmen, B., Ökten, H.H. and Türkmen, E. (2021) Rad-discrete Modules, Bulletin of the Iranian Mathematical Society, 47, 91-100. [8] Türkmen, E. (2013) Rad-⨁-supplemented modules”, Analele Stiintifice ale Universitatii Ovidius Constanta Seria Mathematica, 21 (1), 225-238. [9] Wisbauer, R. (1991) Foundations of Module and Ring Theory (A handbook for study and research)”, Gordon and Breach Science Publishers.