UPPERS TO ZERO IN POLYNOMIAL RINGS WHICH ARE MAXIMAL IDEALS

Title & Authors
UPPERS TO ZERO IN POLYNOMIAL RINGS WHICH ARE MAXIMAL IDEALS
Chang, Gyu Whan;

Abstract
Let D be an integrally closed domain with quotient field K, X be an indeterminate over D, $\small{f=a_0+a_1X+{\cdots}+a_nX^n{\in}D[X]}$ be irreducible in K[X], and $\small{Q_f=fK[X]{\cap}D[X]}$. In this paper, we show that $\small{Q_f}$ is a maximal ideal of D[X] if and only if $\small{(\frac{a_1}{a_0},{\cdots},\frac{a_n}{a_0}){\subseteq}P}$ for all nonzero prime ideals P of D; in this case, $\small{Q_f=\frac{1}{a_0}fD[X]}$. As a corollary, we have that if D is a Krull domain, then D has infinitely many height-one prime ideals if and only if each maximal ideal of D[X] has height $\small{{\geq}2}$.
Keywords
upper to zero;maximal ideal;polynomial ring;G-domain;
Language
English
Cited by
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