covering and intersection conditions for prime ideals
0
조회
0
다운로드
2
인용
0
참고문헌
키워드
초록
Let D be an integral domain, P be a nonzero prime ideal of D, {Pα｜α∈A} be a nonempty set of prime ideals of D, and {Iβ｜β∈B} be a nonempty family of ideals of D with ∩β∈BIβ≠(0). Consider the following conditions: (i) If P ⊆∪α∈APα then P = Pα for some α∈A; (ii) If ∩β∈BIβ⊆P, then Iβ⊆P for some β ∈B. In this paper, we prove that D satisfies (i)⇔D is a generalized weakly factorial domain of dim(D)=1⇔D satisfies (ii)⇔D is a weakly Krull domain of dim(D)=1. We also study the t-operation analogs of (i) and (ii).
Abstract
Let D be an integral domain, P be a nonzero prime ideal of D, {Pα｜α∈A} be a nonempty set of prime ideals of D, and {Iβ｜β∈B} be a nonempty family of ideals of D with ∩β∈BIβ≠(0). Consider the following conditions: (i) If P ⊆∪α∈APα then P = Pα for some α∈A; (ii) If ∩β∈BIβ⊆P, then Iβ⊆P for some β ∈B. In this paper, we prove that D satisfies (i)⇔D is a generalized weakly factorial domain of dim(D)=1⇔D satisfies (ii)⇔D is a weakly Krull domain of dim(D)=1. We also study the t-operation analogs of (i) and (ii).
참고문헌 (12)
- D.D. Anderson, "Finite character representation for integral domains", Bollettino U. M. I., 7(6), pp.613~630 (1992)
- D.F. Anderson, "Generalized weakly factorial domains", Houston J. Math, 29, pp.1~13 (2003)
- G.W. Chang, "Weakly factorial rings with zero divisors", Lecture Notes in Pure and Appl. Math., Marcel Dekker, 220, pp.119~131 (2001)
- R. Gilmer, "Multiplicative Ideal Theory" (1972), Marcel Dekker
- R. Gilmer, "An intersection condition for prime ideals", Lecture Notes in Pure and Appl. Math., Marcel Dekker, 189, pp.327~331 (1997)
- C.J. Hwang, "A note on coverings of prime ideals", Comm. Korean Math. Soc., 14, pp.681~685 (1999)
- C.J. Hwang, "A covering condition for the prime spectrums", East Asian Math. J, 21, pp.61~64 (2005)
- Nv, "J. Algebra", 1989, 151~170
- I. Kaplansky, "Commutative Rings, revised edition" (1974), Univ. of Chicago Press
- C.M. Reis, "A compactness property for prime ideals in Noetherian rings", Proc. Amer. Math. Soc., 25, pp.353~356 (1970)
- W. Smith, "A covering condition for prime ideals", Proc. Amer. Math. Soc, 30, pp.451~452 (1971)
- M. Griffin, "Some results on v-multiplication rings", Canad. Math. J., 19, pp.710~722 (1967)