Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

THE KRONECKER FUNCTION RING OF THE RING D[X]N*

  • Received : 2008.08.15
  • Accepted : 2010.03.06
  • Published : 2010.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

Let D be an integrally closed domain with quotient field K, * be a star operation on D, X, Y be indeterminates over D, $N_*\;=\;\{f\;{\in}\;D[X]|\;(c_D(f))^*\;=\;D\}$ and $R\;=\;D[X]_{N_*}$. Let b be the b-operation on R, and let $*_c$ be the star operation on D defined by $I^{*_c}\;=\;(ID[X]_{N_*})^b\;{\cap}\;K$. Finally, let Kr(R, b) (resp., Kr(D, $*_c$)) be the Kronecker function ring of R (resp., D) with respect to Y (resp., X, Y). In this paper, we show that Kr(R, b) $\subseteq$ Kr(D, $*_c$) and Kr(R, b) is a kfr with respect to K(Y) and X in the notion of [2]. We also prove that Kr(R, b) = Kr(D, $*_c$) if and only if D is a $P{\ast}MD$. As a corollary, we have that if D is not a $P{\ast}MD$, then Kr(R, b) is an example of a kfr with respect to K(Y) and X but not a Kronecker function ring with respect to K(Y) and X.

Keywords

Acknowledgement

Supported by : College of Natural Sciences, University of Incheon

References

  1. D. D. Anderson, Some remarks on the ring R(X), Comment. Math. Univ. St. Paul. 26 (1977/78), no. 2, 137-140.
  2. D. F. Anderson, D. E. Dobbs, and M. Fontana, Characterizing Kronecker function rings, Ann. Univ. Ferrara Sez. VII (N.S.) 36 (1990), 1-13.
  3. G. W. Chang, $\ast$-Noetherian domains and the ring $D[X]_{N_{\ast}}$, J. Algebra 297 (2006), no. 1, 216-233. https://doi.org/10.1016/j.jalgebra.2005.08.020
  4. G. W. Chang, Prufer *-multiplication domains, Nagata rings, and Kronecker function rings, J. Algebra 319 (2008), no. 1, 309-319. https://doi.org/10.1016/j.jalgebra.2007.10.010
  5. G. W. Chang, Overrings of the Kronecker function ring Kr(D,*) of a Prufer *-multiplication domain D, Bull. Korean Math. Soc. 46 (2009), no. 5, 1013-1018. https://doi.org/10.4134/BKMS.2009.46.5.1013
  6. G. W. Chang and J. Park, Star-invertible ideals of integral domains, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 6 (2003), no. 1, 141-150.
  7. M. Fontana and K. A. Loper, An historical overview of Kronecker function rings, Nagata rings, and related star and semistar operations, Multiplicative ideal theory in commutative algebra, 169-187, Springer, New York, 2006.
  8. R. Gilmer, Multiplicative Ideal Theory, Marcel Dekker, Inc., New York, 1972.
  9. E. G. Houston, S. B. Malik, and J. L. Mott, Characterizations of $^*$-multiplication domains, Canad. Math. Bull. 27 (1984), no. 1, 48-52. https://doi.org/10.4153/CMB-1984-007-2
  10. B. G. Kang, Prufer v-multiplication domains and the ring $R[X]_{N_v}$, J. Algebra 123 (1989), no. 1, 151-170. https://doi.org/10.1016/0021-8693(89)90040-9

Cited by

  1. Topological properties of semigroup primes of a commutative ring vol.58, pp.3, 2017, https://doi.org/10.1007/s13366-017-0340-z