• Ouyang, Lunqun
  • Received : 2010.11.08
  • Published : 2012.07.01


Let $R$ be a ring and $nil(R)$ the set of all nilpotent elements of $R$. For a subset $X$ of a ring $R$, we define $N_R(X)=\{a{\in}R{\mid}xa{\in}nil(R)$ for all $x{\in}X$}, which is called a weak annihilator of $X$ in $R$. $A$ ring $R$ is called weak zip provided that for any subset $X$ of $R$, if $N_R(Y){\subseteq}nil(R)$, then there exists a finite subset $Y{\subseteq}X$ such that $N_R(Y){\subseteq}nil(R)$, and a ring $R$ is called weak symmetric if $abc{\in}nil(R){\Rightarrow}acb{\in}nil(R)$ for all a, b, $c{\in}R$. It is shown that a generalized power series ring $[[R^{S,{\leq}}]]$ is weak zip (resp. weak symmetric) if and only if $R$ is weak zip (resp. weak symmetric) under some additional conditions. Also we describe all weak associated primes of the generalized power series ring $[[R^{S,{\leq}}]]$ in terms of all weak associated primes of $R$ in a very straightforward way.


weak annihilator;weak associated prime;generalized power series


  1. D. D. Anderson and V. Camillo, Armendariz rings and Gaussian rings, Comm. Algebra 26 (1998), no. 7, 2265-2272.
  2. S. Annin, Associated primes over skew polynomial rings, Comm. Algebra 30 (2002), no. 5, 2511-2528.
  3. S. Annin, Associated primes over Ore extension rings, J. Algebra Appl. 3 (2004), no. 2, 193-205.
  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. J. Brewer and W. Heinzer, Associated primes of principal ideals, Duke Math. J. 41 (1974), 1-7.
  6. G. A. Elliott and P. Ribenboim, Fields of generalized power series, Arch. Math. (Basel) 54 (1990), no. 4, 365-371.
  7. C. Faith, Associated primes in commutative polynomial rings, Comm. Algebra 28 (2000), no. 8, 3983-3986.
  8. Y. Hirano, On annihilator ideals of a polynomial ring over a noncommutative ring, J. Pure Appl. Algebra 168 (2002), no. 1, 45-52.
  9. C. Y. Hong, N. K. Kim, T. K. Kwak, and Y. Lee, Extensions of zip rings, J. Pure Appl. Algebra 195 (2005), no. 3, 231-242.
  10. J. Lambek, On the representation of modules by sheaves of factor modules, Canad. Math. Bull. 14 (1971), 359-368.
  11. Z. Liu, PF-rings of generalised power series, Bull. Austral. Math. Soc. 57 (1998), no. 3, 427-432.
  12. Z. Liu, Injectivity of modules of generalized inverse polynomials, Comm. Algebra 29 (2001), no. 2, 583-592.
  13. Z. Liu, Special properties of rings of generalized power series, Comm. Algebra 32 (2004), no. 8, 3215-3226.
  14. G. Marks, On 2-primal Ore extensions, Comm. Algebra 29 (2001), no. 5, 2113-2123.
  15. L. Ouyang, Ore extensions of weak zip rings, Glasg. Math. J. 51 (2009), no. 3, 525-537.
  16. L. Ouyang and Y. Chen, On weak symmetric rings, Comm. Algebra 38 (2010), no. 2, 697-713.
  17. P. Ribenboim, Rings of generalized power series: Nilpotent elements, Abh. Math. Sem. Univ. Hamburg 61 (1991), 15-33.
  18. P. Ribenboim, Noetherian rings of generalized power series, J. Pure. Appl. Algebra 79 (1992), no. 3, 293-312.
  19. P. Ribenboim, Semisimple rings and von Neumann regular rings of generalized power series, J. Algebra 198 (1997), no. 2, 327-338.
  20. R. C. Shock, Polynomial rings over finite dimensional rings, Pacific J. Math. 42 (1972), 251-257.

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