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

Abstract

Let D be an integrally closed domain with quotient field K, * be a star operation on D, X, Y be indeterminates over D, <TEX>$N_*\;=\;\{f\;{\in}\;D[X]|\;(c_D(f))^*\;=\;D\}$</TEX> and <TEX>$R\;=\;D[X]_{N_*}$</TEX>. Let b be the b-operation on R, and let <TEX>$*_c$</TEX> be the star operation on D defined by <TEX>$I^{*_c}\;=\;(ID[X]_{N_*})^b\;{\cap}\;K$</TEX>. Finally, let Kr(R, b) (resp., Kr(D, <TEX>$*_c$</TEX>)) 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) <TEX>$\subseteq$</TEX> Kr(D, <TEX>$*_c$</TEX>) 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, <TEX>$*_c$</TEX>) if and only if D is a <TEX>$P{\ast}MD$</TEX>. As a corollary, we have that if D is not a <TEX>$P{\ast}MD$</TEX>, 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.