A property of weakly Krull domains
Authors:
D. D. Anderson and Muhammad Zafrullah
Journal:
Proc. Amer. Math. Soc. 131 (2003), 36893692
MSC (2000):
Primary 13F05
Published electronically:
April 30, 2003
MathSciNet review:
1998175
Fulltext PDF Free Access
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Abstract: We show that a weakly Krull domain satisfies : for every pair there is an such that is invertible. For Noetherian, satisfies if and only if every gradeone prime ideal of is of height one. We also show that a modification of can be used to characterize Noetherian domains that are onedimensional.
 [AMZ]
D.
D. Anderson, J.
L. Mott, and M.
Zafrullah, Finite character representations for integral
domains, Boll. Un. Mat. Ital. B (7) 6 (1992),
no. 3, 613–630 (English, with Italian summary). MR 1191956
(93k:13001)
 [B]
Valentina
Barucci, Mori domains, NonNoetherian commutative ring theory,
Math. Appl., vol. 520, Kluwer Acad. Publ., Dordrecht, 2000,
pp. 57–73. MR 1858157
(2002h:13028)
 [CMZ]
Douglas
Costa, Joe
L. Mott, and Muhammad
Zafrullah, The construction
𝐷+𝑋𝐷_{𝑠}[𝑋], J. Algebra
53 (1978), no. 2, 423–439. MR 0506224
(58 #22046)
 [G]
Robert
Gilmer, Multiplicative ideal theory, Queen’s Papers in
Pure and Applied Mathematics, vol. 90, Queen’s University,
Kingston, ON, 1992. Corrected reprint of the 1972 edition. MR 1204267
(93j:13001)
 [HH]
John
R. Hedstrom and Evan
G. Houston, Some remarks on staroperations, J. Pure Appl.
Algebra 18 (1980), no. 1, 37–44. MR 578564
(81m:13008), http://dx.doi.org/10.1016/00224049(80)901140
 [K]
Irving
Kaplansky, Commutative rings, Revised edition, The University
of Chicago Press, Chicago, Ill.London, 1974. MR 0345945
(49 #10674)
 [MMZ]
Saroj
Malik, Joe
L. Mott, and Muhammad
Zafrullah, On 𝑡invertibility, Comm. Algebra
16 (1988), no. 1, 149–170. MR 921947
(88j:13022), http://dx.doi.org/10.1080/00927878808823566
 [O]
Jack
Ohm, Some counterexamples related to
integral closure in 𝐷[[𝑥]], Trans. Amer. Math. Soc. 122 (1966), 321–333. MR 0202753
(34 #2613), http://dx.doi.org/10.1090/S00029947196602027539
 [P]
Martine
PicavetL’Hermitte, Factorization in some orders with a PID
as integral closure, Algebraic number theory and Diophantine analysis
(Graz, 1998) de Gruyter, Berlin, 2000, pp. 365–390. MR 1770474
(2001h:13001)
 [R]
P.
Ribenboim, Anneaux normaux réels à caractère
fini, Summa Brasil. Math. 3 (1956), 213–253
(French). MR
0097391 (20 #3860)
 [Z]
Muhammad
Zafrullah, Putting 𝑡invertibility to use,
NonNoetherian commutative ring theory, Math. Appl., vol. 520, Kluwer
Acad. Publ., Dordrecht, 2000, pp. 429–457. MR 1858174
(2002g:13009)
 [AMZ]
 D.D. Anderson, J.L. Mott and M. Zafrullah, Finite character representations for integral domains, Boll. Un. Mat. Ital. B (7) 6 (1992), 613630. MR 93k:13001
 [B]
 V. Barucci, Mori domains, NonNoetherian Commutative Ring Theory, Math. Appl., vol. 520, Kluwer Acad. Publ., Dordrecht, 2000, pp. 5773. MR 2002h:13028
 [CMZ]
 D. Costa, J.L. Mott and M. Zafrullah, The construction , J. Algebra 53 (1978), 423439. MR 58:22046
 [G]
 R. Gilmer, Multiplicative Ideal Theory, Queen's Papers in Pure and Appl. Math., vol. 90, Queen's University, Kingston, ON, 1992, corrected reprint of the 1972 edition, Pure Appl. Math., vol. 12, Marcel Dekker, New York. MR 93j:13001
 [HH]
 J.R. Hedstrom and E.G. Houston, Some remarks on staroperations, J. Pure Appl. Algebra 18 (1980), 3744. MR 81m:13008
 [K]
 I. Kaplansky, Commutative Rings, University of Chicago Press, Chicago, IL, 1974, revised edition of the 1970 edition, Allyn and Bacon, Boston. MR 49:10674
 [MMZ]
 S. Malik, J.L. Mott and M. Zafrullah, On invertibility, Comm. Algebra 16 (1988), 149170. MR 88j:13022
 [O]
 J. Ohm, Some counterexamples related to integral closure in , Trans. Amer. Math. Soc. 122 (1966), 321333. MR 34:2613
 [P]
 M. PicavetL'Hermitte, Factorization in some orders with a PID as integral closure, Algebraic Number Theory and Diophantine Analysis (Graz, 1998), de Gruyter, Berlin, 2000, pp. 365390. MR 2001h:13001
 [R]
 P. Ribenboim, Anneaux normaux réels à caractère fini, Summa Brasil. Math. 3 (1956), 213253. MR 20:3860
 [Z]
 M. Zafrullah, Putting invertibility to use, NonNoetherian Commutative Ring Theory, Math. Appl., vol. 520, Kluwer Acad. Publ., Dordrecht, 2000, pp. 429457. MR 2002g:13009
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Additional Information
D. D. Anderson
Affiliation:
Department of Mathematics, University of Iowa, Iowa City, Iowa 52242
Email:
dananderson@uiowa.edu
Muhammad Zafrullah
Affiliation:
Department of Mathematics, Idaho State University, Pocatello, Idaho 832098085
Email:
mzafrullah@usa.net
DOI:
http://dx.doi.org/10.1090/S0002993903070473
PII:
S 00029939(03)070473
Keywords:
Weakly Krull
Received by editor(s):
August 12, 2002
Published electronically:
April 30, 2003
Communicated by:
Bernd Ulrich
Article copyright:
© Copyright 2003 American Mathematical Society
