Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ON JACOBSON AND NIL RADICALS RELATED TO POLYNOMIAL RINGS

  • Kwak, Tai Keun (Department of Mathematics Daejin University) ;
  • Lee, Yang (Department of Mathematics Education Pusan National University) ;
  • Ozcan, A. Cigdem (Department of Mathematics Hacettepe University)
  • Received : 2015.02.26
  • Published : 2016.03.01
Download PDF
⟨ Previous Next ⟩

Abstract

This note is concerned with examining nilradicals and Jacobson radicals of polynomial rings when related factor rings are Armendariz. Especially we elaborate upon a well-known structural property of Armendariz rings, bringing into focus the Armendariz property of factor rings by Jacobson radicals. We show that J(R[x]) = J(R)[x] if and only if J(R) is nil when a given ring R is Armendariz, where J(A) means the Jacobson radical of a ring A. A ring will be called feckly Armendariz if the factor ring by the Jacobson radical is an Armendariz ring. It is shown that the polynomial ring over an Armendariz ring is feckly Armendariz, in spite of Armendariz rings being not feckly Armendariz in general. It is also shown that the feckly Armendariz property does not go up to polynomial rings.

Keywords

Acknowledgement

Supported by : National Research Foundation of Korea(NRF), Pusan National University

References

  1. S. A. Amitsur, A general theory of radicals III, Amer. J. Math. 76 (1954), 126-136. https://doi.org/10.2307/2372404
  2. S. A. Amitsur, Radicals of polynomial rings, Canad. J. Math. 8 (1956), 355-361. https://doi.org/10.4153/CJM-1956-040-9
  3. D. D. Anderson and V. Camillo, Armendariz rings and Gaussian rings, Comm. Algebra 26 (1998), no. 7, 2265-2272. https://doi.org/10.1080/00927879808826274
  4. R. Antoine, Nilpotent elements and Armendariz rings, J. Algebra 319 (2008), no. 8, 3128-3140. https://doi.org/10.1016/j.jalgebra.2008.01.019
  5. R. Antoine, Examples of Armendariz rings, Comm. Algebra 38 (2010), no. 11, 4130-4143. https://doi.org/10.1080/00927870903337968
  6. E. P. Armendariz, A note on extensions of Baer and P.P.-rings, J. Aust. Math. Soc. 18 (1974), 470-473. https://doi.org/10.1017/S1446788700029190
  7. V. Camillo, C. Y. Hong, N. K. Kim, Y. Lee, and P. P. Nielsen, Nilpotent ideals in polynomial and power series rings, Proc. Amer. Math. Soc. 138 (2010), no. 5, 1607-1619. https://doi.org/10.1090/S0002-9939-10-10252-4
  8. J. L. Dorroh, Concerning adjunctions to algebras, Bull. Amer. Math. Soc. 38 (1932), no. 2, 85-88. https://doi.org/10.1090/S0002-9904-1932-05333-2
  9. K. E. Eldridge, Orders for finite noncommutative rings with unity, Amer. Math. Monthly 73 (1966), 512-514.
  10. K. R. Goodearl, Von Neumann Regular Rings, Pitman, London, 1979.
  11. K. R. Goodearl and R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, Cambridge University Press, Cambridge-New York-Port Chester-Melbourne-Sydney, 1989.
  12. J. C. Han, H. K. Kim, and Y. Lee, Armendariz property over prime radicals, J. Korean Math. Soc. 50 (2013), no. 5, 973-989. https://doi.org/10.4134/JKMS.2013.50.5.973
  13. J. C. Han and W. K. Nicholson, Extensions of clean rings, Comm. Algebra 29 (2001), no. 6, 2589-2595. https://doi.org/10.1081/AGB-100002409
  14. Y. Hirano, D. V. Huynh, and J. K. Park, On rings whose prime radical contains all nilpotent elements of index two, Arch. Math. 66 (1996), no. 5, 360-365. https://doi.org/10.1007/BF01781553
  15. S. U. Hwang, Y. C. Jeon, and Y. Lee, Structure and topological conditions of NI rings, J. Algebra 302 (2006), no. 1, 186-199. https://doi.org/10.1016/j.jalgebra.2006.02.032
  16. Y. C. Jeon, H. K. Kim, Y. Lee, and J. S. Yoon, On weak Armendariz rings, Bull. Korean Math. Soc. 46 (2009), no. 1, 135-146. https://doi.org/10.4134/BKMS.2009.46.1.135
  17. D. W. Jung, N. K. Kim, Y. Lee, and S. P. Yang, Nil-Armendariz rings and upper nilradicals, Internat. J. Algebra Comput. 22 (2012), no. 6, 1250059, 13 pp.
  18. N. K. Kim, K. H. Lee, and Y. Lee, Power series rings satisfying a zero divisor property, Comm. Algebra 34 (2006), no. 6, 2205-2218. https://doi.org/10.1080/00927870600549782
  19. N. K. Kim and Y. Lee, On right quasi-duo rings which are ${\pi}$-regular, Bull. Korean Math. Soc. 37 (2000), no. 2, 217-227.
  20. N. K. Kim and Y. Lee, Armendariz rings and reduced rings, J. Algebra 223 (2000), no. 2, 477-488. https://doi.org/10.1006/jabr.1999.8017
  21. J. Lambek, Lectures on Rings and Modules, Blaisdell Publishing Company, Waltham, 1966.
  22. C. Lanski, Nil subrings of Goldie rings are nilpotent, Canad. J. Math. 21 (1969), 904-907. https://doi.org/10.4153/CJM-1969-098-x
  23. T.-K. Lee and T.-L. Wong, On Armendariz rings, Houston J. Math. 29 (2003), no. 3, 583-593.
  24. C. Levitzki, A theorem on polynomial identities, Proc. Amer. Math. Soc. 1 (1950), 331-333. https://doi.org/10.1090/S0002-9939-1950-0036004-9
  25. A. R. Nasr-Isfahani and A. Moussavi, A generalization of reduced rings, J. Algebra Appl. 11 (2012), no. 4, 1250070, 30 pp.
  26. W. K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229 (1977), 269-278. https://doi.org/10.1090/S0002-9947-1977-0439876-2
  27. M. B. Rege and S. Chhawchharia, Armendariz rings, Proc. Japan Acad. Ser. A Math. Sci. 73 (1997), no. 1, 14-17. https://doi.org/10.3792/pjaa.73.14
  28. A. Smoktunowicz, Polynomial rings over nil rings need not be nil, J. Algebra 233 (2000), no. 2, 427-436. https://doi.org/10.1006/jabr.2000.8451
  29. J. Stock, On rings whose projective modules have the exchange property, J. Algebra 103 (1986), no. 2, 437-453. https://doi.org/10.1016/0021-8693(86)90145-6
  30. B. Ungor, O. Gurgun, S. Halicioglu, and A. Harmanci, Feckly reduced rings, Hacettepe J. Math. Stat. 44 (2015), 375-384.
  31. R. B. Warfield, Exchange rings and decompositions of modules, Math. Ann. 199 (1972), 31-36. https://doi.org/10.1007/BF01419573

Cited by

  1. ABELIAN PROPERTY CONCERNING FACTORIZATION MODULO RADICALS vol.24, pp.4, 2016, https://doi.org/10.11568/kjm.2016.24.4.737
  2. Two extensions of right G-semilocal and right N-semilocal rings pp.1793-6829, 2018, https://doi.org/10.1142/S0219498819500129