Pairs of domains where all intermediate domains are Noetherian
Author:
Adrian R. Wadsworth
Journal:
Trans. Amer. Math. Soc. 195 (1974), 201211
MSC:
Primary 13G05
MathSciNet review:
0349665
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: For Noetherian integral domains R and T with is called a Noetherian pair (NP) if every domain A, , is Noetherian. When (Krull dimension) it is shown that the only NP's are those given by the KrullAkizuki Theorem. For , there is another type of NP besides the finite integral extension, namely where . Further, for every NP (R, T) with there is an integral NP extension B of R with . In all known examples B can be chosen to be a finite integral extension of R. For such NP's it is shown that the NP relation is transitive. T may itself be an infinite integral extension R, though, and an example of this is given. It is unknown exactly which infinite integral extensions are NP's.
 [1]
N. Bourbaki, Eléments de mathématique. Fasc. XXX. Algèbre commutative. Chap. 5: Entiers. Chap. 6: Valuations, Actualités Sci. Indust., no. 1308, Hermann, Paris, 1964. MR 33 # 2660. MR 0194450 (33:2660)
 [2]
E. Davis, Integrally closed pairs, Conference on Commutative Algebra, Lecture Notes in Math., vol. 311, Springer, Berlin, 1972, pp. 103106. MR 0335490 (49:271)
 [3]
, Overrings of commutative rings. I. Noetherian overrings, Trans. Amer. Math. Soc. 104 (1962), 5261 MR 25 # 3061. MR 0139629 (25:3061)
 [4]
P. M. Eakin, Jr., The converse to a wellknown theorem on Noetherian rings, Math. Ann. 177 (1968), 278282. MR 37 # 1360. MR 0225767 (37:1360)
 [5]
R. W. Gilmer, Integral domains with Noetherian subrings, Comment. Math. Helv. 45 (1970), 129134. MR 41 #6826. MR 0262216 (41:6826)
 [6]
I. Kaplansky, Commutative rings, Allyn and Bacon, Boston, Mass., 1970. MR 40 #7234. MR 0254021 (40:7234)
 [7]
, Topics in commutative rings. I, mimeographed notes.
 [8]
M. Nagata, Local rings, Interscience Tracts in Pure and Appl. Math., no. 13, Interscience, New York, 1962. MR 27 #5790. MR 0155856 (27:5790)
 [9]
O. Zariski and P. Samuel, Commutative algebra. Vol. II, University Series in Higher Math., Van Nostrand, Princeton, N. J., 1960. MR 22 #11006. MR 0120249 (22:11006)
 [1]
 N. Bourbaki, Eléments de mathématique. Fasc. XXX. Algèbre commutative. Chap. 5: Entiers. Chap. 6: Valuations, Actualités Sci. Indust., no. 1308, Hermann, Paris, 1964. MR 33 # 2660. MR 0194450 (33:2660)
 [2]
 E. Davis, Integrally closed pairs, Conference on Commutative Algebra, Lecture Notes in Math., vol. 311, Springer, Berlin, 1972, pp. 103106. MR 0335490 (49:271)
 [3]
 , Overrings of commutative rings. I. Noetherian overrings, Trans. Amer. Math. Soc. 104 (1962), 5261 MR 25 # 3061. MR 0139629 (25:3061)
 [4]
 P. M. Eakin, Jr., The converse to a wellknown theorem on Noetherian rings, Math. Ann. 177 (1968), 278282. MR 37 # 1360. MR 0225767 (37:1360)
 [5]
 R. W. Gilmer, Integral domains with Noetherian subrings, Comment. Math. Helv. 45 (1970), 129134. MR 41 #6826. MR 0262216 (41:6826)
 [6]
 I. Kaplansky, Commutative rings, Allyn and Bacon, Boston, Mass., 1970. MR 40 #7234. MR 0254021 (40:7234)
 [7]
 , Topics in commutative rings. I, mimeographed notes.
 [8]
 M. Nagata, Local rings, Interscience Tracts in Pure and Appl. Math., no. 13, Interscience, New York, 1962. MR 27 #5790. MR 0155856 (27:5790)
 [9]
 O. Zariski and P. Samuel, Commutative algebra. Vol. II, University Series in Higher Math., Van Nostrand, Princeton, N. J., 1960. MR 22 #11006. MR 0120249 (22:11006)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
13G05
Retrieve articles in all journals
with MSC:
13G05
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197403496652
PII:
S 00029947(1974)03496652
Keywords:
Finitely generated module,
integral extension,
Krull dimension,
KrullAkizuki Theorem,
Noetherian integral domain
Article copyright:
© Copyright 1974
American Mathematical Society
