Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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A STRUCTURE ON COEFFICIENTS OF NILPOTENT POLYNOMIALS

  • Jeon, Young-Cheol (DEPARTMENT OF MATHEMATICS KOREA SCIENCE ACADEMY) ;
  • Lee, Yang (DEPARTMENT OF MATHEMATICS EDUCATION BUSAN NATIONAL UNIVERSITY) ;
  • Ryu, Sung-Ju (DEPARTMENT OF MATHEMATICS BUSAN NATIONAL UNIVERSITY)
  • Received : 2008.04.25
  • Published : 2010.07.01
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Abstract

We observe a structure on the products of coefficients of nilpotent polynomials, introducing the concept of n-semi-Armendariz that is a generalization of Armendariz rings. We first obtain a classification of reduced rings, proving that a ring R is reduced if and only if the n by n upper triangular matrix ring over R is n-semi-Armendariz. It is shown that n-semi-Armendariz rings need not be (n+1)-semi-Armendariz and vice versa. We prove that a ring R is n-semi-Armendariz if and only if so is the polynomial ring over R. We next study interesting properties and useful examples of n-semi-Armendariz rings, constructing various kinds of counterexamples in the process.

Keywords

Acknowledgement

Supported by : National Research Foundation of Korea

References

  1. D. D. Anderson and V. Camillo, Armendariz rings and Gaussian rings, Comm. Algebra 26 (1998), no. 7, 2265–2272.
  2. D. D. Anderson and V. Camillo, Semigroups and rings whose zero products commute, Comm. Algebra 27 (1999), no. 6, 2847-2852. https://doi.org/10.1080/00927879908826596
  3. 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
  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.
  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. 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
  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. J. Lambek, On the representation of modules by sheaves of factor modules, Canad. Math. Bull. 14 (1971), 359-368. https://doi.org/10.4153/CMB-1971-065-1
  11. T.-K. Lee and T.-L. Wong, On Armendariz rings, Houston J. Math. 29 (2003), no. 3, 583–593.
  12. T.-K. Lee and Y. Zhou, Armendariz and reduced rings, Comm. Algebra 32 (2004), no. 6, 2287-2299. https://doi.org/10.1081/AGB-120037221
  13. G. Marks, On 2-primal Ore extensions, Comm. Algebra 29 (2001), no. 5, 2113-2123. https://doi.org/10.1081/AGB-100002173
  14. 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

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