Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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PF-rings of Generalized Power Series

  • Kim, Hwankoo (Department of Information Security, College of Engineering, Hoseo University) ;
  • Kwon, Tae In (Department of Applied Mathematics, Changwon National University)
  • Received : 2006.01.13
  • Published : 2007.03.23
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Abstract

In this paper, we show that if R is a commutative ring with identity and (S, ${\leq}$) is a strictly totally ordered monoid, then the ring [[$R^{S,{\leq}}$]] of generalized power series is a PF-ring if and only if for any two S-indexed subsets A and B of R such that $B{\subseteq}ann_R(|A)$, there exists $c{\in}ann_R(A)$ such that $bc=b$ for all $b{\in}B$, and that for a Noetherian ring R, $[[R^{S,{\leq}}$]] is a PP ring if and only if R is a PP ring.

Keywords

References

  1. H. Al-Ezeh, On some properties of polynomial rings, Int. J. Math. Math. Sci., 10(1987), 311-314. https://doi.org/10.1155/S0161171287000371
  2. H. Al-Ezeh, Two properties of the power series ring, Int. J. Math. Math. Sci., 11(1988), 9-14. https://doi.org/10.1155/S0161171288000031
  3. H. Al-Ezeh, On generalized PF-rings, Math. J. Okayama Univ., 31(1989), 25-29.
  4. G. A. Elliott and P. Ribenboim, Fields of generalized power series, Arch. Math., 54(1990), 365-371. https://doi.org/10.1007/BF01189583
  5. H. Kim, On t-closedness of generalized power series rings, J. Pure Appl. Algebra, 166(2002), 277-284. https://doi.org/10.1016/S0022-4049(01)00020-2
  6. H. Kim and Y. S. Park, Krull domains of generalized power series, J. Algebra, 237(2001), 292-301. https://doi.org/10.1006/jabr.2000.8581
  7. J.-H. Kim, A note on the quotient ring R((X)) of the power series ring R[[X]], J. Korean Math. Soc., 25(1998), 265-271.
  8. Z. Liu, On n-closedness of generalized power series rings over pairs of rings, J. Pure Appl. Algebra, 144(1999), 303-312. https://doi.org/10.1016/S0022-4049(98)00064-4
  9. Z. Liu, Special properties of rings of generalized power series, Comm. Algebra, 32(2004), 3215-3226. https://doi.org/10.1081/AGB-120039287
  10. Z. Liu and J. Ahsan, PP-rings of generalized power series, Acta. Math. Sinica, 16(2000), 573-578. https://doi.org/10.1007/s1011400000884
  11. E. Matlis, The minimal spectrum of a reduced ring, Ill. J. Math., 27(1983), 353-391.
  12. P. Ribenboim, Rings of generalized power series: Nilpotent elements, Abh. Math. Sem. Univ. Hamburg, 61(1991), 15-33. https://doi.org/10.1007/BF02950748
  13. P. Ribenboim, Noetherian rings of generalized power series, J. Pure Appl. Algebra, 79(1992), 293-312. https://doi.org/10.1016/0022-4049(92)90056-L
  14. P. Ribenboim, Rings of generalized power series, II: Units and zero-divisors, J. Algebra, 168(1994), 71-89. https://doi.org/10.1006/jabr.1994.1221
  15. P. Ribenboim, Special properties of generalized power series, J. Algebra, 173(1995), 566-586. https://doi.org/10.1006/jabr.1995.1103
  16. J. J. Rotman, "An Introduction to Homological Algebra", Academic Press, San Diego, 1993.