Skip to main content
SpringerLink
Log in

On \(\mathcal {Z}\)-Reversible Rings

  • Scientific Research Paper
  • Published:
Proceedings of the National Academy of Sciences, India Section A: Physical Sciences Aims and scope Submit manuscript

Abstract

We introduce the idea of a right \(\mathcal {Z}\)-reversible ring. The class of \(\mathcal {Z}\)-reversible rings contains the class of Dedekind finite rings and it contained in the classes of central reversible and right \(\mathcal {Z}\)-reduced rings. In support, we provide many examples of \(\mathcal {Z}\)-reversible and non \(\mathcal {Z}\)-reversible rings. We study some characterizations and extensions of \(\mathcal {Z}\)-reversible rings along with several properties.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Lam TY (1999) Lectures on modules and rings. Graduate Texts in Mathematics. Springer, Berlin

    Book  MATH  Google Scholar 

  2. Cohn PM (1999) Reversible rings. Bull Lond Math Soc 31(6):641–648

    Article  MathSciNet  MATH  Google Scholar 

  3. Kose H, Ungor B, Halicioglu S, Harmanci A (2014) A generalization of reversible rings. Iran J Sci Technol Trans A Sci 38(1):43–48

    MathSciNet  Google Scholar 

  4. Calci MB, Chen H, Halicioglu S, Harmanci A (2017) Reversibility of rings with respect to the Jacobson radical. Mediterr J Math 14(3):137

    Article  MathSciNet  MATH  Google Scholar 

  5. Lam TY (2001) A first course in noncommutative rings, 2nd edn. Graduate Texts in Mathematics. Springer, New York

    Book  MATH  Google Scholar 

  6. Lambek J (1971) On the representation of modules by sheaves of factor modules. Can Math Bull 14(3):359–368

    Article  MathSciNet  MATH  Google Scholar 

  7. Marks G (2002) Reversible and symmetric rings. J Pure Appl Algebra 174(3):311–318

    Article  MathSciNet  MATH  Google Scholar 

  8. Nejadzadeh A, Amini A, Amini B, Sharif H (2018) Rings in which nilpotent elements are right singular. Bull Iran Math Soc 44(5):1217–1226

    Article  MathSciNet  MATH  Google Scholar 

  9. Agayev N, Harmanci A, Halicioglu S (2009) Extended Armendariz rings. Algebras Groups Geom 26(4):343–354

    MathSciNet  MATH  Google Scholar 

  10. Chen H, Gurgun O, Halicioglu S, Harmanci A (2016) Rings in which nilpotents belong to Jacobson radical. An Stiint Univ Al I Cuza Iasi Mat (NS) T 2:595–606

    MathSciNet  MATH  Google Scholar 

  11. Amini A, Amini B, Nejadzadeh A, Sharif H (2018) Singular clean rings. J Korean Math Soc 55(5):1143–1156

    MathSciNet  MATH  Google Scholar 

  12. Nicholson WK (1977) Lifting idempotents and exchange rings. Trans Am Math Soc 229:269–278

    Article  MathSciNet  MATH  Google Scholar 

  13. Lam TY (2003) Exercises in classical ring theory, 2nd edn. Problem Books in Mathematics. Springer, New York

    MATH  Google Scholar 

  14. Wei J (2008) Certain rings whose simple singular modules are nil-injective. Turkish J Math 32(4):393–408

    MathSciNet  MATH  Google Scholar 

  15. Rege MB, Chhawchharia S (1997) Armendariz rings. Proc Jpn Acad Ser A Math Sci 73(1):14–17

    Article  MathSciNet  MATH  Google Scholar 

  16. Zhao L, Zhu X, Gu Q (2013) Reflexive rings and their extensions. Math Slovaca 63(3):417–430

    Article  MathSciNet  MATH  Google Scholar 

  17. Nicholson WK, Campos ES (2004) Rings with the dual of the isomorphism theorem. J Algebra 271(1):391–406

    Article  MathSciNet  MATH  Google Scholar 

  18. Lee CI, Park SY (2018) When nilpotents are contained in Jacobson radicals. J Korean Math Soc 55(5):1193–1205

    MathSciNet  MATH  Google Scholar 

  19. Camillo V, Nielsen PP (2008) McCoy rings and zero-divisors. J Pure Appl Algebra 212(3):599–615

    Article  MathSciNet  MATH  Google Scholar 

  20. Yue R, Ming C (1983) On quasi-frobeniusean and Artinian rings. Publications de l’institut mathematique Nouvelle series 33(47):239–245

    MathSciNet  MATH  Google Scholar 

  21. Vaserstein LN (1984) Bass’s first stable range condition. J Pure Appl Algebra 34(2–3):319–330

    Article  MathSciNet  MATH  Google Scholar 

  22. Nejadzadeh A, Amini A, Amini B, Sharif H (2020) \({\cal{Z}}\)-Armendariz rings and modules. Vietnam J Math 48(1):131–143

    Article  MathSciNet  MATH  Google Scholar 

  23. Ferrero M, Puczylowski ER (1998) The singular ideal and radicals. J Aust Math Soc 64(2):195–209

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We are thankful to reviewer(s) for their valuable comments to improve this paper. The research of the second author was supported by the Junior Research Fellowship grant, University Grants Commission, India (UGC Ref. No.: 1211/(CSIR-UGC NET DEC. 2018)).

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Allahabad, Prayagraj, 211002, India

    Avanish Kumar Chaturvedi & Nirbhay Kumar

Authors
  1. Avanish Kumar Chaturvedi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nirbhay Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Avanish Kumar Chaturvedi.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chaturvedi, A.K., Kumar, N. On \(\mathcal {Z}\)-Reversible Rings. Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. 92, 555–562 (2022). https://doi.org/10.1007/s40010-022-00770-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40010-022-00770-3

Keywords

Mathematics Subject Classification

Search

Navigation