Journal of Sciences, Islamic Republic of Iran

On Projection Invariant Rickart Modules

Document Type : Original Paper

Authors

1 1 Department of Pure Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran, Islamic Republic of Iran

2 Bursa Uludag University, Faculty of Arts and Science, Department of Mathematics, 16059 Gorukle, Bursa/TURKEY

Abstract

This study examined π-Rickart modules, a module-theoretic analog of π-Rickart rings, from the perspective of their endomorphism rings. It is shown that π-Rickart conditions are located between π-e. Baer and p.q.-Baer conditions, and it is established that the corresponding endomorphism ring possesses the appropriate π-Rickart property. Besides, the notion of π-e.AIP modules is presented. Furthermore, connections to the aforementioned concepts of π-Rickart, endo-AIP, and π-e.AIP modules are examined.

Keywords

Main Subjects

  1. Kaplansky I. Rings of Operators New York: Benjamin; 1965.
  2. Clark WE. Twisted matrix units semigroup algebras. Duke Math J. 1967; 34(3): 417–23.
  3. Maeda S. On a ring whose principal right ideals generated by idempotents form a lattice. J Sci Hiroshima Univ Ser A. 1960; 24: 509-25.
  4. Armendariz EP. A note on extensions of Baer and P.P.-rings. J Austra Math Soc. 1947; 18: 470-73.
  5. Berberian K. Baer *-Rings New York-Berlin: Springer Berlin, Heidelberg.
  6. Bergman GM. Hereditary commutative rings and centres of hereditary rings. Proc London Math Soc. 1971; 23(4): 214-36.
  7. Evans M. On commutative P.P. rings. Pacific J Math. 1972; 41(3): 687–97.
  8. Hattori A. A foundation of torsion theory for modules over general rings. Nagoya Math J. 1960; 17: 147-58.
  9. Jondrup S. p.p. rings and finitely generated flat ideals. Proc Ame. Math Soc. 1971; 28(2): 431-35.
  10. Rizvi ST, Roman CS. Baer and quasi-Baer modules. Comm Algebra. 2004; 32(1): 103-23.
  11. Lee G. Principally quasi-Baer modules and their generalizations. Comm Algebra. 2019; 47(10): 4077-94.
  12. Lee G, Rizvi ST, Roman CS. Rickart Modules. Comm Algebra. 2010; 38(11): 4005-27.
  13. Lee G, Rizvi ST, Roman CS. 𝔏-Rickart Modules. Comm Algebra. 2015; 43(5): 2124-51.
  14. Birkenmeier GF, Kara Y, Tercan A. π-Baer rings. J Algebra Appl. 2018; 17(2).
  15. Birkenmeier GF, Kara Y, Tercan A. π-endo Baer modules. Comm Algebra. 2020; 48(3): 1132-49.
  16. Birkenmeier GF, Kara Y, Tercan A. π-Rickart rings. J Algebra Appl. 2021; 20(8).
  17. Birkenmeier GF, Kara Y, Tercan A. Primitive and prime rings with s.Baer or related modules. J Algebra Appl. 2024; 23(03).
  18. Birkenmeier GF, Kilic N, Mutlu , Tastan , Tercan , Yasar. Connections between Baer annihilator conditions and extending conditions for nearrings and rings. J Algebra Appl. 2025.
  19. Kara Y. Weak π-Rickart rings. Comm Algebra. 2025;: 1-12.
  20. KESKIN TÜTÜNCÜ D. A Study on the π-Dual Rickart Modules. J. Iran. Math. Soc. 2023 july; 4(2): 235-45.
  21. Maleki Y, Moussavi A, Kara Y. On π-Endo.Rickart Modules. Iran J Sci. 2024.
  22. Birkenmeier GF, Kim JY, Park JK. Principally quasi-Baer rings. Comm Algebra. 2001; 29(2): 638-60.
  23. Lorenz M. K0 of Skew Group Rings and Simple Noetherian Rings without idempotents. J Lond Math Soc. 1985; 32(5): 41-50.
  24. Birkenmeier GF, Tercan A, Yucel CC. The extending condition relative to sets of submodules. Commun Algebra. 2014; 42(2): 764-78.
  25. Liu Q, Ouyang B, Wu T. Principally quasi-Baer modules. J Math Res Appl. 2009; 29(5): 823-30.
  26. Huh C, Lee Y, Smoktunowicz A. Armendariz rings and semicommutative rings. Commun Algebra. 2002; 30(2): 751-61.
  27. Lee G, Rizvi ST, Roman CS. Dual Rickart Modules. Comm Algebra. 2011; 39(11): 4036-58.
  28. Tominaga H. On s-unital rings. Mathematical Journal of Okayama University. 1976; 18(2): 117-34.
  29. Amirzadeh Dana P, Moussavi A. Modules in which the annihilator of a fully invariant submodule is pure. Comm Algebra. 2020; 48(11): 4875-88.
  30. Wolfson KG. Baer rings of endomorphisms. Math Ann. 1961; 143(1): 19-28.
  31. Majidinya A, Moussavi A, Paykan K. Rings in which the annihilator of an ideal is pure. Algebra Colloq. 2015; 22(01): 947-68.
  32. Birkenmeier GF, Park JK, Rizvi ST. Extensions of Rings and Modules New York: Birkhauser; 2013.
  33. Ozcan AC, Harmanci A, Smith PF. Duo modules. Glasg Math J. 2006; 48(03): 533-45.
  34. Chatters AW, Khuri SM. Endomorphism rings of modules over nonsingular CS rings. J London Math Soc. 1980; 21(2): 434-44.
  35. Birkenmeier GF, Kara Y, Tercan A. Primitive and prime rings with s.Baer or related modules. J Algebra Appl. 2024; 23(03).

 

Journal of Sciences, Islamic Republic of Iran
Volume 35, Issue 4
October 2024
Pages 321-328