Hostname: page-component-77c89778f8-m42fx Total loading time: 0 Render date: 2024-07-18T09:22:54.987Z Has data issue: false hasContentIssue false
  • English
  • Français

Quasi-projectivity over domains

Published online by Cambridge University Press:  17 April 2009

Dmitri Alexeev
Dmitri Alexeev
Affiliation:
Department of MathematicsTulane UniversityNew Orleans LA 70118United States of America e-mail: alexeev@math.tulane.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let R be an integral domain with quotient field Q. We investigate quasi- and Q-projective ideals, and properties of domains all ideals of which are quasi-projective. It is shown that the so-called l½-generated ideals are quasi-projective, moreover, projective. A module M is quasi-projective if and only if, for a projective ideal P of R, the tensor product M ⊗RP is quasi-projective. Domains whose all ideals are quasi-projective are characterised as almost maximal Prüfer domains. Q is quasi-projective if and only if every proper submodule of Q is complete in its R-topology.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 60 , Issue 1 , August 1999 , pp. 129 - 135
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Anderson, F. and Fuller, K., Rings and categories of modules, (second edition) (Springer-Verlag, Berlin, Heidelberg, New York, 1992).Google Scholar
[2]Brandal, W., ‘Almost maximal integral domains and finitely generated modules’, Trans. Amer. Math. Soc. 183 (1973), 203–222.Google Scholar
[3]Herrmann, P., ‘Self-projective modules over valuation rings’, Arch. Math. 43 (1984), 332–339.Google Scholar
[4]Lantz, D.C. and Martin, M.B., ‘Strongly two-generated ideals’, Comm. Alg. 16 (1988), 1759–1777.CrossRefGoogle Scholar
[5]Matlis, E., ‘Injective modules over Prüfer rings’, Nagoya Math. 15 (1959), 57–69.Google Scholar
[6]Matlis, E., ‘Cotorsion modules’, Mem. Amer. Math. Soc. 49 (1964).Google Scholar
[7]Miyashita, Y., ‘Quasi-projective modules, perfect modules and a theorem for modular lattices’, J. Fac. Sci. Hokkaido Univ. Ser. I 19 (1966), 86–110.Google Scholar
[8]Olberding, B., ‘Almost maximal Prüfer domains’ (to appear).Google Scholar
[9]Rangaswamy, K. M. and Vanaja, N., ‘Quasi projectives in Abelian and module categories’, Pacific J. Math. 43 (1972), 221–238.Google Scholar