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Year 2019, Volume: 48 Issue: 6, 1744 - 1760, 08.12.2019
https://doi.org/10.15672/HJMS.2018.634

Abstract

References

  • [1] S.A. Amitsur, Radicals of polynomial rings, Canad. J. Math. 8, 355-361, 1956.
  • [2] R. Antoine, Nilpotent elements and Armendariz rings, J. Algebra 319, 3128-3140, 2008.
  • [3] G.F. Birkenmeier, H.E. Heatherly, and E.K. Lee, Completely prime ideals and associated radicals, in: Proc. Biennial Ohio State-Denison Conference 1992, edited by S.K. Jain and S.T. Rizvi, World Scientific, Singapore-New Jersey-London-Hong Kong, 102-129, 1993.
  • [4] W. Chen, On linearly weak Armendariz rings, J. Pure Appl. Algebra 219, 1122-1130, 2015.
  • [5] E.-K. Cho, T.K. Kwak, Y. Lee, Z. Piao, and Y. Seo, A structure of noncentral idempotents, Bull. Korean Math. Soc. 55, 25-40, 2018.
  • [6] Y. Chun, Y. C. Jeon, S. Kang, K. N. Lee, Y. Lee, A concept unifying the Armendariz and NI conditions, Bull. Korean Math. Soc. 48, 115-127, 2011.
  • [7] E.H. Feller, Properties of primary noncommutative rings, Trans. Amer. Math. Soc. 89, 79-91, 1958.
  • [8] K.R. Goodearl, Von Neumann Regular Rings, Pitman, London 1979.
  • [9] K.R. Goodearl and Jr. R.B.Warfield, An Introduction to Noncommutative Noetherian Rings, Cambridge University Press, Cambridge-New York-Port Chester-Melbourne- Sydney, 1989.
  • [10] C. Huh, H.K. Kim, and Y. Lee, p.p. rings and generalized p.p. rings, J. Pure Appl. Algebra 167, 37-52, 2002.
  • [11] S.U. Hwang, Y.C. Jeon, and Y. Lee, Structure and topological conditions of NI rings, J. Algebra 302, 186-199, 2006.
  • [12] H.K. Kim, T.K. Kwak, Y. Lee, and Y. Seo, Corrigendum Insertion of units at zero products [J. Algebra Appl. Vol. 17, 1850043 (20 pp), 2018], J. Algebra Appl. Vol. 17, 1892002, 2pp, 2018.
  • [13] N.K. Kim and Y. Lee, Armendariz rings and reduced rings, J. Algebra 223, 477-488, 2018.
  • [14] N.K. Kim, Y. Lee, and Y. Seo, Structure of idempotents in rings without identity, J. Korean Math. Soc. 51, 751-771, 2014.

Quasi-normality of idempotents on nilpotents

Year 2019, Volume: 48 Issue: 6, 1744 - 1760, 08.12.2019
https://doi.org/10.15672/HJMS.2018.634

Abstract

We study the structure of idempotents in non-Abelian rings, concerning a ring property near to the normality of idempotents on the set of nilpotents. We call a ring with such property right idempotent-quasi-normalizing on nilpotents (simply, right IQNN), and study the structure of right IQNN rings in relation with matrix rings, polynomial ring, and factor rings, by which we extend the class of right IQNN rings. It is proved that the class of IQNN rings contains the $2$ by $2$ full matrix rings over fields and the upper triangular matrix rings over reduced rings. It is shown that given any countable field $K$, there exists a semiprime IQNN algebra $R$ over $K$ such that the polynomial ring $R[x]$ over $R$ is IQNN but not NI, and the upper nilradical of $R[x]$ is zero.

References

  • [1] S.A. Amitsur, Radicals of polynomial rings, Canad. J. Math. 8, 355-361, 1956.
  • [2] R. Antoine, Nilpotent elements and Armendariz rings, J. Algebra 319, 3128-3140, 2008.
  • [3] G.F. Birkenmeier, H.E. Heatherly, and E.K. Lee, Completely prime ideals and associated radicals, in: Proc. Biennial Ohio State-Denison Conference 1992, edited by S.K. Jain and S.T. Rizvi, World Scientific, Singapore-New Jersey-London-Hong Kong, 102-129, 1993.
  • [4] W. Chen, On linearly weak Armendariz rings, J. Pure Appl. Algebra 219, 1122-1130, 2015.
  • [5] E.-K. Cho, T.K. Kwak, Y. Lee, Z. Piao, and Y. Seo, A structure of noncentral idempotents, Bull. Korean Math. Soc. 55, 25-40, 2018.
  • [6] Y. Chun, Y. C. Jeon, S. Kang, K. N. Lee, Y. Lee, A concept unifying the Armendariz and NI conditions, Bull. Korean Math. Soc. 48, 115-127, 2011.
  • [7] E.H. Feller, Properties of primary noncommutative rings, Trans. Amer. Math. Soc. 89, 79-91, 1958.
  • [8] K.R. Goodearl, Von Neumann Regular Rings, Pitman, London 1979.
  • [9] K.R. Goodearl and Jr. R.B.Warfield, An Introduction to Noncommutative Noetherian Rings, Cambridge University Press, Cambridge-New York-Port Chester-Melbourne- Sydney, 1989.
  • [10] C. Huh, H.K. Kim, and Y. Lee, p.p. rings and generalized p.p. rings, J. Pure Appl. Algebra 167, 37-52, 2002.
  • [11] S.U. Hwang, Y.C. Jeon, and Y. Lee, Structure and topological conditions of NI rings, J. Algebra 302, 186-199, 2006.
  • [12] H.K. Kim, T.K. Kwak, Y. Lee, and Y. Seo, Corrigendum Insertion of units at zero products [J. Algebra Appl. Vol. 17, 1850043 (20 pp), 2018], J. Algebra Appl. Vol. 17, 1892002, 2pp, 2018.
  • [13] N.K. Kim and Y. Lee, Armendariz rings and reduced rings, J. Algebra 223, 477-488, 2018.
  • [14] N.K. Kim, Y. Lee, and Y. Seo, Structure of idempotents in rings without identity, J. Korean Math. Soc. 51, 751-771, 2014.
There are 14 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Tai Keun Kwak 0000-0001-6316-8650

Seung İck Lee This is me 0000-0002-9612-7844

Yang Lee This is me 0000-0002-7572-5191

Publication Date December 8, 2019
Published in Issue Year 2019 Volume: 48 Issue: 6

Cite

APA Kwak, T. K., Lee, S. İ., & Lee, Y. (2019). Quasi-normality of idempotents on nilpotents. Hacettepe Journal of Mathematics and Statistics, 48(6), 1744-1760. https://doi.org/10.15672/HJMS.2018.634
AMA Kwak TK, Lee Sİ, Lee Y. Quasi-normality of idempotents on nilpotents. Hacettepe Journal of Mathematics and Statistics. December 2019;48(6):1744-1760. doi:10.15672/HJMS.2018.634
Chicago Kwak, Tai Keun, Seung İck Lee, and Yang Lee. “Quasi-Normality of Idempotents on Nilpotents”. Hacettepe Journal of Mathematics and Statistics 48, no. 6 (December 2019): 1744-60. https://doi.org/10.15672/HJMS.2018.634.
EndNote Kwak TK, Lee Sİ, Lee Y (December 1, 2019) Quasi-normality of idempotents on nilpotents. Hacettepe Journal of Mathematics and Statistics 48 6 1744–1760.
IEEE T. K. Kwak, S. İ. Lee, and Y. Lee, “Quasi-normality of idempotents on nilpotents”, Hacettepe Journal of Mathematics and Statistics, vol. 48, no. 6, pp. 1744–1760, 2019, doi: 10.15672/HJMS.2018.634.
ISNAD Kwak, Tai Keun et al. “Quasi-Normality of Idempotents on Nilpotents”. Hacettepe Journal of Mathematics and Statistics 48/6 (December 2019), 1744-1760. https://doi.org/10.15672/HJMS.2018.634.
JAMA Kwak TK, Lee Sİ, Lee Y. Quasi-normality of idempotents on nilpotents. Hacettepe Journal of Mathematics and Statistics. 2019;48:1744–1760.
MLA Kwak, Tai Keun et al. “Quasi-Normality of Idempotents on Nilpotents”. Hacettepe Journal of Mathematics and Statistics, vol. 48, no. 6, 2019, pp. 1744-60, doi:10.15672/HJMS.2018.634.
Vancouver Kwak TK, Lee Sİ, Lee Y. Quasi-normality of idempotents on nilpotents. Hacettepe Journal of Mathematics and Statistics. 2019;48(6):1744-60.