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Strongly nil *-clean rings

Year 2017, , 155 - 164, 11.01.2017
https://doi.org/10.13069/jacodesmath.284954

Abstract

A $*$-ring $R$ is called {\em strongly nil $*$-clean} if every element of $R$ is the sum of a
projection and a nilpotent element that commute with each other.
In this paper we investigate some properties of strongly nil
$*$-rings and prove that $R$ is a strongly nil $*$-clean ring if
and only if every idempotent in $R$ is a projection, $R$ is
periodic, and $R/J(R)$ is Boolean. We also prove that a $*$-ring
$R$ is
commutative, strongly nil $*$-clean and every primary ideal is maximal if and only if every element of $R$ is a projection.

References

  • [1] A. Badawi, On abelian $\pi$–regular rings, Comm. Algebra 25(4) (1997) 1009–1021.
  • [2] S. K. Berberian, Baer *–Rings, Springer-Verlag, Heidelberg, London, New York, 2011.
  • [3] M. Chacron, On a theorem of Herstein, Canad. J. Math. 21 (1969) 1348–1353.
  • [4] H. Chen, On strongly J–clean rings, Comm. Algebra 38(10) (2010) 3790–3804.
  • [5] H. Chen, Rings Related Stable Range Conditions, Series in Algebra 11, World Scientific, Hackensack, NJ, 2011.
  • [6] H. Chen, A. Harmancı A. Ç. Özcan, Strongly J–clean rings with involutions, Ring theory and its applications, Contemp. Math. 609 (2014) 33–44.
  • [7] A. J. Diesl, Nil clean rings, J. Algebra 383 (2013) 197–211.
  • [8] A. L. Foster, The theory of Boolean–like rings, Trans. Amer. Math. Soc. 59 (1946) 166–187.
  • [9] Y. Hirano, H. Tominaga, A. Yaqub, On rings in which every element is uniquely expressable as a sum of a nilpotent element and a certain potent element, Math. J. Okayama Univ. 30 (1988) 33–40.
  • [10] C. Li, Y. Zhou, On strongly *–clean rings, J. Algebra Appl. 10(6) (2011) 1363–1370.
  • [11] V. Swaminathan, Submaximal ideals in a Boolean–like rings, Math. Sem. Notes Kobe Univ. 10(2) (1982) 529–542.
  • [12] L. Vaš, *–Clean rings; some clean and almost clean Baer *–rings and von Neumann algebras, J. Algebra 324(12) (2010) 3388–3400.
Year 2017, , 155 - 164, 11.01.2017
https://doi.org/10.13069/jacodesmath.284954

Abstract

References

  • [1] A. Badawi, On abelian $\pi$–regular rings, Comm. Algebra 25(4) (1997) 1009–1021.
  • [2] S. K. Berberian, Baer *–Rings, Springer-Verlag, Heidelberg, London, New York, 2011.
  • [3] M. Chacron, On a theorem of Herstein, Canad. J. Math. 21 (1969) 1348–1353.
  • [4] H. Chen, On strongly J–clean rings, Comm. Algebra 38(10) (2010) 3790–3804.
  • [5] H. Chen, Rings Related Stable Range Conditions, Series in Algebra 11, World Scientific, Hackensack, NJ, 2011.
  • [6] H. Chen, A. Harmancı A. Ç. Özcan, Strongly J–clean rings with involutions, Ring theory and its applications, Contemp. Math. 609 (2014) 33–44.
  • [7] A. J. Diesl, Nil clean rings, J. Algebra 383 (2013) 197–211.
  • [8] A. L. Foster, The theory of Boolean–like rings, Trans. Amer. Math. Soc. 59 (1946) 166–187.
  • [9] Y. Hirano, H. Tominaga, A. Yaqub, On rings in which every element is uniquely expressable as a sum of a nilpotent element and a certain potent element, Math. J. Okayama Univ. 30 (1988) 33–40.
  • [10] C. Li, Y. Zhou, On strongly *–clean rings, J. Algebra Appl. 10(6) (2011) 1363–1370.
  • [11] V. Swaminathan, Submaximal ideals in a Boolean–like rings, Math. Sem. Notes Kobe Univ. 10(2) (1982) 529–542.
  • [12] L. Vaš, *–Clean rings; some clean and almost clean Baer *–rings and von Neumann algebras, J. Algebra 324(12) (2010) 3388–3400.
There are 12 citations in total.

Details

Subjects Engineering
Journal Section Articles
Authors

Abdullah Harmanci

Huanyin Chen

A. Cigdem Ozcan

Publication Date January 11, 2017
Published in Issue Year 2017

Cite

APA Harmanci, A., Chen, H., & Ozcan, A. C. (2017). Strongly nil *-clean rings. Journal of Algebra Combinatorics Discrete Structures and Applications, 4(2 (Special Issue: Noncommutative rings and their applications), 155-164. https://doi.org/10.13069/jacodesmath.284954
AMA Harmanci A, Chen H, Ozcan AC. Strongly nil *-clean rings. Journal of Algebra Combinatorics Discrete Structures and Applications. May 2017;4(2 (Special Issue: Noncommutative rings and their applications):155-164. doi:10.13069/jacodesmath.284954
Chicago Harmanci, Abdullah, Huanyin Chen, and A. Cigdem Ozcan. “Strongly Nil *-Clean Rings”. Journal of Algebra Combinatorics Discrete Structures and Applications 4, no. 2 (Special Issue: Noncommutative rings and their applications) (May 2017): 155-64. https://doi.org/10.13069/jacodesmath.284954.
EndNote Harmanci A, Chen H, Ozcan AC (May 1, 2017) Strongly nil *-clean rings. Journal of Algebra Combinatorics Discrete Structures and Applications 4 2 (Special Issue: Noncommutative rings and their applications) 155–164.
IEEE A. Harmanci, H. Chen, and A. C. Ozcan, “Strongly nil *-clean rings”, Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 4, no. 2 (Special Issue: Noncommutative rings and their applications), pp. 155–164, 2017, doi: 10.13069/jacodesmath.284954.
ISNAD Harmanci, Abdullah et al. “Strongly Nil *-Clean Rings”. Journal of Algebra Combinatorics Discrete Structures and Applications 4/2 (Special Issue: Noncommutative rings and their applications) (May 2017), 155-164. https://doi.org/10.13069/jacodesmath.284954.
JAMA Harmanci A, Chen H, Ozcan AC. Strongly nil *-clean rings. Journal of Algebra Combinatorics Discrete Structures and Applications. 2017;4:155–164.
MLA Harmanci, Abdullah et al. “Strongly Nil *-Clean Rings”. Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 4, no. 2 (Special Issue: Noncommutative rings and their applications), 2017, pp. 155-64, doi:10.13069/jacodesmath.284954.
Vancouver Harmanci A, Chen H, Ozcan AC. Strongly nil *-clean rings. Journal of Algebra Combinatorics Discrete Structures and Applications. 2017;4(2 (Special Issue: Noncommutative rings and their applications):155-64.