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

Chang, Gyu-Whan

  • Received : 2008.08.15
  • Accepted : 2010.03.06
  • Published : 2010.09.30

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

(e.a.b.) star operation;Kronecker function ring (KFR);kfr;Nagata ring;P*MD

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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

Acknowledgement

Supported by : College of Natural Sciences, University of Incheon