Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ON SEMI-ARMENDARIZ MATRIX RINGS

  • KOZLOWSKI, KAMIL (Faculty of Computer Science Bialystok University of Technology) ;
  • MAZUREK, RYSZARD (Faculty of Computer Science Bialystok University of Technology)
  • Received : 2014.09.12
  • Published : 2015.06.01
Download PDF
⟨ Previous Next ⟩

Abstract

Given a positive integer n, a ring R is said to be n-semi-Armendariz if whenever $f^n=0$ for a polynomial f in one indeterminate over R, then the product (possibly with repetitions) of any n coefficients of f is equal to zero. A ring R is said to be semi-Armendariz if R is n-semi-Armendariz for every positive integer n. Semi-Armendariz rings are a generalization of Armendariz rings. We characterize when certain important matrix rings are n-semi-Armendariz, generalizing some results of Jeon, Lee and Ryu from their paper (J. Korean Math. Soc. 47 (2010), 719-733), and we answer a problem left open in that paper.

Keywords

References

  1. 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
  2. E. P. Armendariz, A note on extensions of Baer and P.P.-rings, J. Austral. Math. Soc. 18 (1974), 470-473. https://doi.org/10.1017/S1446788700029190
  3. B. J. Gardner and R. Wiegandt, Radical theory of rings, Monographs and Textbooks in Pure and Applied Mathematics, vol. 261, Marcel Dekker, Inc., New York, 2004.
  4. Y. Hirano, On annihilator ideals of a polynomial ring over a noncommutative ring, J. Pure Appl. Algebra 168 (2002), no. 1, 45-52. https://doi.org/10.1016/S0022-4049(01)00053-6
  5. C. Huh, H. K. Kim, and Y. Lee, p.p. rings and generalized p.p. rings, J. Pure Appl. Algebra 167 (2002), no. 1, 37-52. https://doi.org/10.1016/S0022-4049(01)00149-9
  6. C. Huh, Y. Lee, and A. Smoktunowicz, Armendariz rings and semicommutative rings, Comm. Algebra 30 (2002), no. 2, 751-761. https://doi.org/10.1081/AGB-120013179
  7. Y. C. Jeon, Y. Lee, and S. J. Ryu, A structure on coefficients of nilpotent polynomials, J. Korean Math. Soc. 47 (2010), no. 4, 719-733. https://doi.org/10.4134/JKMS.2010.47.4.719
  8. 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
  9. 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
  10. T. Y. Lam, A first Course in Noncommutative Rings, Graduate Texts in Math., vol. 131, Springer-Verlag, Berlin-Heidelberg-New York 1991.
  11. T.-K. Lee and T.-L. Wong, On Armendariz rings, Houston J. Math. 29 (2003), no. 3, 583-593.
  12. Z. Liu and R. Zhao, On weak Armendariz rings, Comm. Algebra 34 (2006), no. 7, 2607-2616. https://doi.org/10.1080/00927870600651398
  13. G. Marks, R. Mazurek, and M. Ziembowski, A unified approach to various generalizations of Armendariz rings, Bull. Aust. Math. Soc. 81 (2010), no. 3, 361-397. https://doi.org/10.1017/S0004972709001178
  14. R. Mazurek and M. Ziembowski, Right Gaussian rings and skew power series rings, J. Algebra 330 (2011), no. 1, 130-146. https://doi.org/10.1016/j.jalgebra.2010.11.014
  15. R. Mazurek and M. Ziembowski, On a characterization of distributive rings via saturations and its applications to Armendariz and Gaussian rings, Rev. Mat. Iberoam. 30 (2014), no. 3, 1073-1088. https://doi.org/10.4171/RMI/807