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http://dx.doi.org/10.5666/KMJ.2011.51.3.339

Kaplansky-type Theorems, II  

Chang, Gyu-Whan (Department of Mathematics, University of Incheon)
Kim, Hwan-Koo (Department of Information Security, Hoseo University)
Publication Information
Kyungpook Mathematical Journal / v.51, no.3, 2011 , pp. 339-344 More about this Journal
Abstract
Let D be an integral domain with quotient field K, X be an indeterminate over D, and D[X] be the polynomial ring over D. A prime ideal Q of D[X] is called an upper to zero in D[X] if Q = fK[X] ${\cap}$ D[X] for some f ${\in}$ D[X]. In this paper, we study integral domains D such that every upper to zero in D[X] contains a prime element (resp., a primary element, a t-invertible primary ideal, an invertible primary ideal).
Keywords
Kaplansky theorem; upper to zero in D[X]; prime (primary) element;
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Times Cited By KSCI : 1  (Citation Analysis)
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1 A. Bouvier, Le groupe des classes d'un anneau integre, 107eme Congres des Societes Savantes, Brest, 1982, fasc. IV, 85-92.
2 S. Gabelli, On divisorial ideals in polynomial rings over Mori domains, Comm. Algebra, 15(1987), 2349-2370.   DOI
3 R. Gilmer, Multiplicative Ideal Theory, Marcel Dekker, New York, 1972.
4 E. Houston and M. Zafrullah, On t-invertibility, II, Comm. Algebra, 17(1989), 1955-1969.   DOI   ScienceOn
5 B. G. Kang, Prufer $\upsilon$-multiplication domains and the ring $R[X]_{Nv}$ , J. Algebra, 123(1989), 151-170.   DOI
6 B. G. Kang, On the converse of a well-known fact about Krull domains, J. Algebra, 124(1989), 284-299.   DOI
7 I. Kaplansky, Commutative Rings, rev. ed., Univ. of Chicago, Chicago, 1974.
8 H. Kim, Kaplansky-type theorems, Kyungpook Math. J., 40(2000), 9-16.   과학기술학회마을
9 R. Lewin, Almost generalized GCD-domains, Lecture Notes in Pure and Appl. Math., Marcel Dekker, 189(1997), 371-382.
10 M. Zafrullah, A general theory of almost factoriality, Manuscripta Math., 51(1985), 29-62.   DOI
11 D. D. Anderson, T. Dumitrescu, and M. Zafrullah, Quasi-Schreier domains, II, Comm. Algebra, 35(2007), 2096-2104.   DOI   ScienceOn
12 D. D. Anderson and M. Zafrullah, Almost Bezout domains, J. Algebra, 142(1991), 285-309.   DOI
13 D. D. Anderson and M. Zafrullah, On a theorem of Kaplansky, Boll. Un. Mat. Ital. A (7), 8(1994), 397-402.
14 D. F. Anderson and G. W. Chang, Almost splitting sets in integral domains II, J. Pure Appl. Algebra, 208(2007), 351-359.   DOI   ScienceOn
15 D. F. Anderson, G. W. Chang, and J. Park, Generalized weakly factorial domains, Houston J. Math., 29(2003), 1-13.