Taiwanese Journal of Mathematics

Primitive Submodules, Co-semisimple and Regular Modules

Mauricio Medina-Bárcenas and A. Çiğdem Özcan

Full-text: Open access

Abstract

In this paper, primitive submodules are defined and various properties of them are investigated. Some characterizations of co-semisimple modules are given and several conditions under which co-semisimple and regular modules coincide are discussed.

Article information

Source
Taiwanese J. Math., Volume 22, Number 3 (2018), 545-565.

Dates
Received: 27 August 2017
Revised: 20 November 2017
Accepted: 21 November 2017
First available in Project Euclid: 29 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1511924485

Digital Object Identifier
doi:10.11650/tjm/171102

Mathematical Reviews number (MathSciNet)
MR3807325

Zentralblatt MATH identifier
06965385

Subjects
Primary: 16D40: Free, projective, and flat modules and ideals [See also 19A13] 16D60: Simple and semisimple modules, primitive rings and ideals 16D90: Module categories [See also 16Gxx, 16S90]; module theory in a category-theoretic context; Morita equivalence and duality 16P20: Artinian rings and modules 16D80: Other classes of modules and ideals [See also 16G50]

Keywords
primitive right ideals co-semisimple modules regular modules and rings fully bounded modules artinian modules

Citation

Medina-Bárcenas, Mauricio; Özcan, A. Çiğdem. Primitive Submodules, Co-semisimple and Regular Modules. Taiwanese J. Math. 22 (2018), no. 3, 545--565. doi:10.11650/tjm/171102. https://projecteuclid.org/euclid.twjm/1511924485


Export citation

References

  • F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Graduate Texts in Mathematics 13, Springer-Verlag, New York-Heidelberg, 1974.
  • G. Baccella, On flat factor rings and fully right idempotent rings, Ann. Univ. Ferrara Sez. VII (N.S.) 26 (1980), 125–141.
  • ––––, Von Neumann regularity of $V$-rings with Artinian primitive factor rings, Proc. Amer. Math. Soc. 103 (1988), no. 3, 747–749.
  • J. A. Beachy, $M$-injective modules and prime $M$-ideals, Comm. Algebra 30 (2002), no. 10, 4649–4676.
  • L. Bican, P. Jambor, T. Kepka and P. Němec, Prime and coprime modules, Fund. Math. 107 (1980), no. 1, 33–45.
  • J. Castro Pérez and J. Ríos Montes, Prime submodules and local Gabriel correspondence in $\sigma[M]$, Comm. Algebra 40 (2012), no. 1, 213–232.
  • ––––, FBN modules, Comm. Algebra 40 (2012), no. 12, 4604–4616.
  • ––––, Krull dimension and classical Krull dimension of modules, Comm. Algebra 42 (2014), no. 7, 3183–3204.
  • J. Castro Pérez, M. Medina Bárcenas, J. Ríos Montes and A. Zaldívar Corichi, On semiprime Goldie modules, Comm. Algebra 44 (2016), no. 11, 4749–4768.
  • ––––, On the structure of Goldie modules, Comm. Algebra, Accepted.
  • A. W. Chatters and C. R. Hajarnavis, Rings with Chain Conditions, Research Notes in Mathematics 44, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1980.
  • K. R. Fuller, Relative projectivity and injectivity classes determined by simple modules, J. London Math. Soc. (2) 5 (1972), 423–431.
  • K. R. Goodearl, Von Neumann Regular Rings, Second edition, Robert E. Krieger, Malabar, FL, 1991.
  • K. R. Goodearl and R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, London Mathematical Society Student Texts 16, Cambridge University Press, Cambridge, 1989.
  • A. Haghany, M. Mazrooei and M. R. Vedadi, Bounded and fully bounded modules, Bull. Aust. Math. Soc. 84 (2011), no. 3, 433–440.
  • S. M. Khuri, Endomorphism rings and lattice isomorphisms, J. Algebra 56 (1979), no. 2, 401–408.
  • H. Lee, Right FBN rings and bounded modules, Comm. Algebra 16 (1988), no. 5, 977–987.
  • A. Ç. Özcan, A. Harmanci and P. F. Smith, Duo modules, Glasg. Math. J. 48 (2006), no. 3, 533–545.
  • F. Raggi, J. Ríos, H. Rincón, R. Fernández-Alonso and C. Signoret, Prime and irreducible preradicals, J. Algebra Appl. 4 (2005), no. 4, 451–466.
  • ––––, Semiprime preradicals, Comm. Algebra 37 (2009), no. 8, 2811–2822.
  • V. S. Ramamurthi, A note on regular modules, Bull. Austral. Math. Soc. 11 (1974), 359–364.
  • N. V. Sanh, S. Asawasamrit, K. F. U. Ahmed and L. P. Thao, On prime and semiprime Goldie modules, Asian-Eur. J. Math. 4 (2011), no. 2, 321–334.
  • P. F. Smith, Modules with many homomorphisms, J. Pure Appl. Algebra 197 (2005), no. 1-3, 305–321.
  • A. A. Tuganbaev, Semidistributive Modules and Rings, Mathematics and its Applications 4449, Kluwer Academic Publishers, Dordrecht, 1998.
  • ––––, Semiregular, weakly regular, and $\pi$-regular rings, Algebra 16. J. Math. Sci. (New York) 109 (2002), no. 3, 1509–1588.
  • ––––, Rings Close to Regular, Mathematics and its Applications 545, Kluwer Academic Publishers, Dordrecht, 2002.
  • J. E. Van den Berg and R. Wisbauer, Modules whose hereditary pretorsion classes are closed under products, J. Pure Appl. Algebra 209 (2007), no. 1, 215–221.
  • R. Wisbauer, Generalized co-semisimple modules, Comm. Algebra 18 (1990), no. 12, 4235–4253.
  • ––––, Foundations of Module and Ring Theory: A handbook for study and research, Algebra, Logic and Applications 3, Gordon and Breach Science Publishers, Philadelphia, PA, 1991.
  • ––––, Modules and Algebras: bimodule structure and group actions on algebras, Pitman Monoggraphs and Surveys in Pure and Applied Mathematics 81, Longman, Harlow, 1996.
  • H. P. Yu, On quasi-duo rings, Glasgow Math. J. 37 (1995), no. 1, 21–31.
  • J. Zelmanowitz, Regular modules, Trans. Amer. Math. Soc. 163 (1972), 341–355.