UPPERS TO ZERO IN POLYNOMIAL RINGS WHICH ARE MAXIMAL IDEALS
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Let D be an integrally closed domain with quotient field K, X be an indeterminate over D, f = a0 + a1X + · · · + anXn ∈ D[X] be irreducible in K[X], and Qf = fK[X] ∩ D[X]. In this paper, we show that Qf is a maximal ideal of D[X] if and only if ( a1/a0 , . . . , an/a0 ) ⊆ P for all nonzero prime ideals P of D; in this case, Qf = 1/a0 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 ≥ 2.
Abstract
Let D be an integrally closed domain with quotient field K, X be an indeterminate over D, f = a0 + a1X + · · · + anXn ∈ D[X] be irreducible in K[X], and Qf = fK[X] ∩ D[X]. In this paper, we show that Qf is a maximal ideal of D[X] if and only if ( a1/a0 , . . . , an/a0 ) ⊆ P for all nonzero prime ideals P of D; in this case, Qf = 1/a0 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 ≥ 2.
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