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. Let D be an integrally closed domain with quotient eld K,X be an indeterminate over D, f = a 0 + a 1 X+ ··· + a n X n ∈ D[X] beirreducible in K[X], and Q f = fK[X] ∩ D[X]. In this paper, we showthat Q f is a maximal ideal of D[X] if and only if ( a 1 a 0 ,..., a n a 0 ) ⊆ P for allnonzero prime ideals P of D; in this case, Q f = 1a 0 fD[X]. As a corollary,we have that if D is a Krull domain, then D has innitely many height-one prime ideals if and only if each maximal ideal of D[X] has height≥ 2. 1. IntroductionLet D be an integral domain with quotient eld K, D¯ the integral closureof D in K, X an indeterminate over D, and D[X] the polynomial ring over D.We say that a nonzero prime ideal Q of D[X] is an upper to zero in D[X] ifQ∩D = (0); so each upper to zero in D[X] has height-one. Clearly, if Q is anupper to zero in D[X], then Q = fK[X]∩ D[X] for some f ∈ D[X] which isirreducible in K[X].Let X 1 ,...,X n be indeterminates over D. It is known that the intersectionof the nonzero prime ideals of D is zero if and only if M ∩ D 6= (0) for allmaximal ideals M of D[X

UPPERS TO ZERO IN POLYNOMIAL RINGS WHICH ARE MAXIMAL IDEALS