Skip to main content
SpringerLink
Log in

Some New Results on Skew Triangular Matrix Rings with Constant Diagonal

  • Published:
Vietnam Journal of Mathematics Aims and scope Submit manuscript

Abstract

This paper concerns mainly on various ring properties of some subrings of a skew triangular matrix ring. Necessary and sufficient conditions are obtained for a skew triangular matrix ring with constant diagonal over a ring to satisfy a certain ring property which is among being local, semilocal, semiperfect, semiregular, left quasi-duo, clean, (weakly) nil clean, exchange, uniquely (nil) clean, Hermite ring, stably finite, semiregular and I-ring. It is also proved that the projective-free property of a ring preserves by a skew triangular matrix ring with constant diagonal.

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. Breaz, S., Danchev, P., Zhou, Y.: Rings in which every element is either a sum or a difference of a nilpotent and an idempotent. J. Algebra Appl. 15, 1650148 (2016). 11 pages

  2. Boxun, Z.: Homological Algebra. Scientific Press, Beijing (1988)

    Google Scholar 

  3. Camillo, V.P., Yu, H.-P.: Exchange rings, units, and idempotents. Commun. Algebra 22, 4737–4749 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen, J., Yang, X., Zhou, Y.: On strongly clean matrix and triangular matrix rings. Commun. Algebra 34, 3659–3674 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cohn, P.M.: Free Rings and their Relations, 2Nd edn. London Mathematical Society Monographs, vol. 19. Academic Press, London–New York (1985)

  6. Diesl, A.J.: Classes of Strongly Clean Rings. Ph.D. Dissertation. University of California, Berkeley (2006)

    Google Scholar 

  7. Diesl, A.J.: Nil clean rings. J. Algebra 383, 197–211 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Evans, E.G.: Krull-schmidt and cancellation over local rings. Pac. J. Math. 46, 115–121 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  9. Habibi, M., Moussavi, A.: Special properties of a skew triangular matrix ring with constant diagonal. Asian-Eur. J. Math. 8, 1550021 (2015). 10 pages

    Article  MathSciNet  MATH  Google Scholar 

  10. Habibi, M., Moussavi, A., Alhevaz, A.: On skew triangular matrix rings. Algebra Colloq. 22, 271–280 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kanwar, P., Leroy, A., Matczuk, J.: Idempotents in ring extensions. J. Algebra 389, 128–136 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lam, T.Y.: A First Course in Noncommutative Rings Graduate Texts in Mathematics, vol. 131. Springer-Verlag, New York (1991)

  13. Lam, T.Y., Dugas, A.S.: Quasi-duo rings and stable range descent. J. Pure Appl. Algebra 195, 243–259 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Leroy, A., Matczuk, J., Puczyowski, E.R.: Quasi-duo skew polynomial rings. J. Pure Appl. Algebra 212, 1951–1959 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Montgomery, S.: Von Neumann finiteness of tensor products of algebras. Commun. Algebra 11, 595–610 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  16. Nasr-Isfahani, A.R., Moussavi, A.: On a quotient of polynomial rings. Commun. Algebra 38, 567–575 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Nasr-Isfahani, A.R.: On skew triangular matrix rings. Commun. Algebra 39, 4461–4469 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Nasr-Isfahani, A.R.: On a quotient of skew polynomial rings. Commun. Algebra 41, 4520–4533 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Nicholson, W.K.: I-rings. Trans. Am. Math. Soc. 207, 361–373 (1975)

    MATH  Google Scholar 

  20. Nicholson, W.K.: Semiregular modules and rings. Can. J. Math. 28, 1105–1120 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  21. Nicholson, W.K.: Lifting idempotents and exchange rings. Tran. Am. Math. Soc. 229, 269–278 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  22. Nicholson, W.K., Zhou, Y.: Rings in which elements are uniquely the sum of an idempotent and a unit. Glasg. Math. J. 46, 227–236 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  23. Vámos, P.: 2-good rings. Q. J. Math. 56, 417–430 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wang, Y., Ren, Y.: 2-good rings and their extensions. Bull. Korean Math. Soc. 50, 1711–1723 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Warfield, R.B.: Exchange rings and decompositions of modules. Math. Ann. 199, 31–36 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  26. Warfield, R.B.: Serial rings and finitely presented modules. J. Algebra 37, 187–222 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  27. Yu, H.-P.: On quasi-duo rings. Glasg. Math. J. 37, 21–31 (1995)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The author would like to express their deep gratitude to the referee for a very careful reading of the article, and many valuable comments, which have greatly improved the presentation of the article.

Author information

Authors and Affiliations

  1. Department of Mathematics, Garmsar Branch, Islamic Azad University, Garmsar, Iran

    Kamal Paykan

Authors
  1. Kamal Paykan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kamal Paykan.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Paykan, K. Some New Results on Skew Triangular Matrix Rings with Constant Diagonal. Vietnam J. Math. 45, 575–584 (2017). https://doi.org/10.1007/s10013-016-0229-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10013-016-0229-4

Keywords

Mathematics Subject Classification (2010)

Search

Navigation