Kyungpook Mathematical Journal

  • 제60권4호
  • /
  • Pages.711-721
  • /
  • 2020
  • /
  • 1225-6951(pISSN)
  • /
  • 0454-8124(eISSN)
경북대학교 자연과학대학 수학과 (Department of Mathematics, Kyungpook National University)
권호 표지
DOI QR코드

DOI QR Code

On n-Amitsur Rings

  • Ochirbat, Baatar (Department of Mathematics, Mongolian University of Science and Technology) ;
  • Mendes, Deolinda I.C. (Department of Mathematics, University of Beira Interior) ;
  • Tumurbat, Sodnomkhorloo (Department of Mathematics, National University of Mongolia and School of Applied Science and Technology)
  • 투고 : 2019.08.25
  • 심사 : 2020.07.21
  • 발행 : 2020.12.31
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

The concepts of an Amitsur ring and a hereditary Amitsur ring, which were introduced and studied by S. Tumurbat in a recent paper, are generalized. For a positive integer n, a ring A is said to be an n-Amitsur ring if γ(A[Xn]) = (γ(A[Xn]) ∩ A)[Xn] for all radicals γ, where A[Xn] is the polynomial ring over A in n commuting indeterminates. If a ring A satisfies the above equation for all hereditary radicals γ, then A is said to be a hereditary n-Amitsur ring. Characterizations and examples of these rings are provided. Moreover, new radicals associated with n-Amitsur rings are introduced and studied. One of these is a special radical and its semisimple class is polynomially extensible.

키워드

과제정보

The first and third authors were partly supported by the Science and Technology Fund of Mongolia, Grant No. Shuss2017/64. The second author was partly supported by the research project: Grant UID/MAT/00212/2019 - financed by FEDER through the - Programa Operacional Factores de Competividade, FCT - Fundação para a Ciência e Teconologia.

참고문헌

  1. N. Divinsky and A. Sulinski, Kurosh radicals of rings with operators, Canad. J. Math., 17(1965), 278-280. https://doi.org/10.4153/CJM-1965-027-6
  2. H. France-Jackson, T. Khulan and S. Tumurbat, On α-like radicals of rings, Bull. Aust. Math. Soc., 88(2)(2013), 331-339. https://doi.org/10.1017/S0004972712000895
  3. B. J. Gardner, J. Krempa and R. Wiegandt, Open problems in radical theory, Algebra Discrete Math., (3)(2007), 15-17.
  4. B. J. Gardner and R. Wiegandt, Radical theory of rings, Marcel Dekker, New York, 2004.
  5. B. J. Gardner and L. Zhian, Small and large radical classes, Comm. Algebra, 20(1992), 2533-2551.
  6. J. Krempa, On radical properties of polynomial rings, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys., 20(1972), 545-548.
  7. N. V. Loi and R. Wiegandt, On the Amitsur property of radicals, Algebra Discrete Math., (3)(2006), 92-100.
  8. N. R. McConnell and T. Stokes, Radical ideals of radically simple rings and their extensions, Theory of radicals, 185-196, Colloquia Mathematica Societatis Janos Bolyai 61, North-Holland, Amsterdam, 1993.
  9. S. Tumurbat, On Amitsur rings, Quaest. Math., 42(5)(2019), 665-672. https://doi.org/10.2989/16073606.2018.1480542
  10. S. Tumurbat, D. Dayantsolmon and T. Khulan, Notes on chain rings and radicals, Kyungpook Math. J., 58(2018), 473-479. https://doi.org/10.5666/KMJ.2018.58.3.473
  11. S. Tumurbat and R. Wiegandt, Radicals of polynomial rings, Soochow J. Math., 29(4)(2003), 425-434.
  12. S. Tumurbat and R. Wiegandt. On radicals with Amitsur property, Comm. Algebra, 32(3)(2004), 1219-1227. https://doi.org/10.1081/AGB-120027976
  13. S. Tumurbat and R. Wisbauer, Radicals with the α-Amitsur property, J. Algebra Appl., 7(3)(2008), 347-361. https://doi.org/10.1142/S0219498808002837
  14. R. Wiegandt, Radical and semisimple classes of rings, Queen's Papers in Pure and Applied Mathematics 37, Queen's University, Kingston Ontario, 1974.