Uppers to zero in polynomial rings which are maximal ideals
Bull. Korean Math. Soc. 2015 Vol. 52, No. 2, 525-530
https://doi.org/10.4134/BKMS.2015.52.2.525
Published online March 31, 2015
Gyu Whan Chang
Incheon National University
Abstract : Let $D$ be an integrally closed domain with quotient field $K$, $X$ be an indeterminate over $D$, $f = a_0 + a_1X + \cdots + a_nX^n \in D[X]$ be irreducible in $K[X]$, and $Q_f = fK[X] \cap D[X]$. In this paper, we show that $Q_f$ is a maximal ideal of $D[X]$ if and only if $(\frac{a_1}{a_0}, \dots , \frac{a_n}{a_0}) \subseteq P$ for all nonzero prime ideals $P$ of $D$; in this case, $Q_f = \frac{1}{a_0}fD[X]$. 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 $\geq 2$.
Keywords : upper to zero, maximal ideal, polynomial ring, G-domain
MSC numbers : 13A15, 13B25, 13G05
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