• Ouyang, Lunqun (Department of Mathematics Hunan University of Science and Technology) ;
  • Liu, Jinwang (Department of Mathematics and Applied Mathematics Huaihua University)
  • Received : 2012.10.25
  • Published : 2013.07.31


We in this note introduce a concept, so called nil generalized power serieswise Armendariz ring, that is a generalization of both S-Armendariz rings and nil power serieswise Armendariz rings. We first observe the basic properties of nil generalized power serieswise Armendariz rings, constructing typical examples. We next study the relationship between the nilpotent property of R and that of the generalized power series ring [[$R^{S,{\leq}}$]] whenever R is nil generalized power serieswise Armendariz.


  1. D. D. Anderson and V. Camillo, Armendariz rings and Gaussian rings, Comm. Algebra 26 (1998), no. 7, 2265-2272.
  2. R. Antoine, Nilpotent elements and Armendariz rings, J. Algebra 319 (2008), no. 8, 3128-3140.
  3. E. P. Armendariz, A note on extensions of Baer and p.p.-rings, J. Austral. Math. Soc. 18 (1974), 470-473.
  4. J. A. Beachy and W. D. Blair, Rings whose faithful left ideals are cofaithful, Pacific J. Math. 58 (1975), no. 1, 1-13.
  5. H. Cartan and E. Eilenberg, Homological Algebra, Princeton Landmarks inMathematics, originally published in 1956, Princeton: Princeton University Press, 1956.
  6. C. Faith, Rings with zero intersection property on annihilators: zip rings, Publ. Mat. 33 (1989), no. 2, 329-332.
  7. S. Hizem, A note on nil power serieswise Armendariz rings, Rendiconti del Circolo Mathematico di Palermo 59 (2010), no. 1, 87-99.
  8. 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.
  9. T. Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics, Springer-Verlag, Berlin, 1991.
  10. Z. K. Liu, Special properties of rings of generalized power series, Comm. Algebra 32 (2004), no. 8, 3215-3226.
  11. Z. K. Liu, On weak Armendariz rings, Comm. Algebra 34 (2006), no. 7, 2607-2616.
  12. L. Ouyang, Ore extensions of weak zip rings, Glasg. Math. J. 51 (2009), no. 3, 525-537.
  13. M. B. Rege and S. Chhawchharia, Armendariz rings, Proc. Japan Acad. Ser. A Math. Sci. 73 (1997), no. 1, 14-17.
  14. P. Ribenboim, Rings of generalized power series: Nilpotent elements, Abh. Math. Sem. Univ. Hamburg 61 (1991), 15-33.
  15. P. Ribenboim, Noetherian rings of generalized power series, J. Pure Appl. Algebra 79 (1992), no. 3, 293-312.
  16. P. Ribenboim, Semisimple rings and von Neumann regular rings of generalized power series, J. Algebra 198 (1997), no. 2, 327-338.