Finiteness in prime ideals in rings of global dimension two

Sheng, Hu

doi:10.1090/S0002-9939-1973-0327739-4

Finiteness in prime ideals in rings of global dimension two
HTML articles powered by AMS MathViewer

by Hu Sheng PDF

Proc. Amer. Math. Soc. 41 (1973), 363-369 Request permission

Abstract:

Let $A$ be a commutative ring with identity. The main result states conditions that ensure the finiteness of prime ideals in a coherent ring $A$ of global dimension two. Precisely, any ideal containing two noncomparable prime ideals is finitely generated. As a corollary it follows that a Krull domain of global dimension two is noetherian. Another corollary is that if $A$ is not semihereditary it contains a finitely generated maximal ideal.

References

Maurice Auslander, On the dimension of modules and algebras. III. Global dimension, Nagoya Math. J. 9 (1955), 67–77. MR 74406

Eléments de mathématique

Algèbre commutative

Modules plats

Localisation

36

Stephen U. Chase, Direct products of modules, Trans. Amer. Math. Soc. 97 (1960), 457–473. MR 120260, DOI 10.1090/S0002-9947-1960-0120260-3
Joel M. Cohen, A note on homological dimension, J. Algebra 11 (1969), 483–487. MR 237609, DOI 10.1016/0021-8693(69)90087-8
Christian U. Jensen, Some remarks on a change of rings theorem, Math. Z. 106 (1968), 395–401. MR 238891, DOI 10.1007/BF01115089
Irving Kaplansky, Commutative rings, Allyn and Bacon, Inc., Boston, Mass., 1970. MR 0254021
Irving Kaplansky, Fields and rings, University of Chicago Press, Chicago, Ill.-London, 1969. MR 0269449
K. R. Nagarajan, Groups acting on Noetherian rings, Nieuw Arch. Wisk. (3) 16 (1968), 25–29. MR 229628
P. Samuel, Lectures on unique factorization domains, Tata Institute of Fundamental Research Lectures on Mathematics, No. 30, Tata Institute of Fundamental Research, Bombay, 1964. Notes by M. Pavman Murthy. MR 0214579
Wolmer V. Vasconcelos, On finitely generated flat modules, Trans. Amer. Math. Soc. 138 (1969), 505–512. MR 238839, DOI 10.1090/S0002-9947-1969-0238839-5
Wolmer V. Vasconcelos, The local rings of global dimension two, Proc. Amer. Math. Soc. 35 (1972), 381–386. MR 308115, DOI 10.1090/S0002-9939-1972-0308115-6
Wolmer V. Vasconcelos, Rings on global dimension two, Conference on Commutative Algebra (Univ. Kansas, Lawrence, Kan., 1972), Lecture Notes in Math., Vol. 311, Springer, Berlin, 1973, pp. 243–251. MR 0340239
Wolmer V. Vasconcelos, Finiteness in projective ideals, J. Algebra 25 (1973), 269–278. MR 314828, DOI 10.1016/0021-8693(73)90045-8

Coherence of one polynomial ring