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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.
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참고문헌 (12)
참고문헌 신청
B. G. Kang / 1989 / Pr¨ufer v-multiplication domains and the ring R[X]Nv / J. Algebra 123 (1) : 151 ~ 170
D. F. Anderson / 2007 / Almost splitting sets in integral domains. II / J. Pure Appl. Algebra 208 (1) : 351 ~ 359
E. Artin / 1951 / A note on finite ring extensions / J. Math. Soc. Japan 3 : 74 ~ 77
E. Houston / 1989 / On t-invertibility. II / Comm. Algebra 17 (8) : 1955 ~ 1969
F. Zanello / 2004 / When are there infinitely many irreducible elements in a principal ideal domain? / Amer. Math. Monthly 111 (2) : 150 ~ 152
D. F. Anderson / 2007 / Almost splitting sets in integral domains. II / J. Pure Appl. Algebra 208 (1) : 351 ~ 359
E. Artin / 1951 / A note on finite ring extensions / J. Math. Soc. Japan 3 : 74 ~ 77
E. Houston / 1989 / On t-invertibility. II / Comm. Algebra 17 (8) : 1955 ~ 1969
F. Zanello / 2004 / When are there infinitely many irreducible elements in a principal ideal domain? / Amer. Math. Monthly 111 (2) : 150 ~ 152
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