Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Coherence of one polynomial ring


Author: Wolmer V. Vasconcelos
Journal: Proc. Amer. Math. Soc. 41 (1973), 449-456
MSC: Primary 13F20
DOI: https://doi.org/10.1090/S0002-9939-1973-0325608-7
MathSciNet review: 0325608
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The main result is that $ A[x]$, the polynomial ring in one variable over the domain $ A$ of global dimension two, is coherent. As an application, the quotient rings of $ A[x]$ which have finite global dimension are examined.


References [Enhancements On Off] (What's this?)

  • [1] N. Bourbaki, Eléments de mathématique. Fasc. XXVII. Algèbre commutative. Chap. 1: Modules plats. Chap. 2: Localisation, Actualités Sci. Indust., no. 1290, Hermann, Paris, 1961. MR 36 #146.
  • [2] L. Burch, On ideals of finite homological dimension in local rings, Proc. Cambridge Philos. Soc. 64 (1968), 941-948. MR 37 #5208. MR 0229634 (37:5208)
  • [3] S. U. Chase, Direct product of modules, Trans. Amer. Math. Soc. 97 (1960), 457-473. MR 22 #11017. MR 0120260 (22:11017)
  • [4] C. U. Jensen, On homological dimensions of rings with countably generated ideals, Math. Scand. 18 (1966), 97-105. MR 34 #7611. MR 0207796 (34:7611)
  • [5] I. Kaplansky, Commutative rings, Allyn and Bacon, Boston, Mass., 1970. MR 40 #7234. MR 0254021 (40:7234)
  • [6] I. Kaplansky, Fields and rings, University of Chicago Press, Chicago, Ill., 1969. MR 42 #4345. MR 0269449 (42:4345)
  • [7] M. Raynaud and L. Gruson, Critères de platitude et de projectivité, Invent. Math. 13 (1971), 1-89. MR 0308104 (46:7219)
  • [8] W. V. Vasconcelos, The local rings of global dimension two, Proc. Amer. Math. Soc. 35 (1972), 381-386. MR 0308115 (46:7230)
  • [9] -, Rings of global dimension two, Proc. Comm. Algebra Conf. (Lawrence, Kansas, 1972), Lecture Notes in Math., no. 311, Springer-Verlag, Berlin, 1973. MR 0340239 (49:4994)
  • [10] -, Finiteness in projective ideals, J. Algebra 25 (1973), 269-278. MR 0314828 (47:3378)
  • [11] W. V. Vasconcelos and A. Simis, Projective modules over $ R[x],R$ valuation ring, are free, Notices Amer. Math. Soc. 18 (1971), 805. Abstract #71T-A178.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 13F20

Retrieve articles in all journals with MSC: 13F20


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0325608-7
Keywords: Projective dimension, global dimension, coherent ring, G.C.D. domain
Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society