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)

 
 

 

Modules over semihereditary Bezout rings


Author: Thomas S. Shores
Journal: Proc. Amer. Math. Soc. 46 (1974), 211-213
MSC: Primary 13F10
DOI: https://doi.org/10.1090/S0002-9939-1974-0349663-4
MathSciNet review: 0349663
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that every commutative semihereditary Bezout ring of Krull dimension at most one is an elementary divisor ring. A consequence is that the ring of polynomials in one indeterminate over a von Neumann regular ring is an elementary divisor ring.


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

  • [B] N. Bourbaki, Eléments de mathématique. Fasc. XXVII. Algèbre commutative. Chap. 1: Modules plats, Actualités Sci. Indust., no. 1290, Hermann, Paris, 1961. MR 36 #146.
  • [C] S. U. Chase, Direct products of modules, Trans. Amer. Math. Soc. 97 (1960), 457-473. MR 22 #11017. MR 0120260 (22:11017)
  • [GH] L. Gillman and M. Henriksen, Some remarks about elementary divisor rings, Trans. Amer. Math. Soc. 82 (1956), 362-365. MR 18, 9. MR 0078979 (18:9c)
  • [GP] R. Gilmer and T. Parker, Semigroup rings as Prüferrings (preprint).
  • [H] M. Henriksen, Some remarks on elementary divisor rings. II, Michigan Math. J. 3 (1955/56), 159-163. MR 19, 1155. MR 0092772 (19:1155i)
  • [K$ _{1}$] I. Kaplansky, Elementary divisors and modules, Trans. Amer. Math. Soc. 66 (1949), 464-491. MR 11, 155. MR 0031470 (11:155b)
  • [K$ _{2}$] -, Infinite abelian groups, rev. ed., Univ. of Michigan Press, Ann Arbor, Mich., 1969. MR 38 #2208. MR 0233887 (38:2208)
  • [LLS] M. Larsen, W. Lewis and T. Shores, Elementary divisor rings and finitely presented modules, Trans. Amer. Math. Soc. 187 (1974), 231-248. MR 0335499 (49:280)
  • [M] P. J. McCarthy, The ring of polynomials over a von Neumann regular ring, Proc. Amer. Math. Soc. 39 (1973), 253-254. MR 0316496 (47:5043)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 13F10

Retrieve articles in all journals with MSC: 13F10


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1974-0349663-4
Keywords: Commutative von Neumann regular ring, polynomial ring, semihereditary ring, Hermite ring, Bezout ring, elementary divisor ring
Article copyright: © Copyright 1974 American Mathematical Society

American Mathematical Society