Hostname: page-component-76fb5796d-vfjqv Total loading time: 0 Render date: 2024-04-26T16:06:07.143Z Has data issue: false hasContentIssue false
  • English
  • Français

Torsionfree modules and classes of orders

Published online by Cambridge University Press:  17 April 2009

William H. Gustafson
William H. Gustafson
Affiliation:
Department of Mathematics, Indiana University, Indiana, USA.
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.

It is shown how torsionfree modules can be used to characterize certain important classes of orders over Dedekind rings. In particular, we show that an order is Gorenstein if and only if each of its lattices can be embedded as a pure sublattice of a free lattice. We also show that an order is hereditary if and only if the tensor product of any of its right lattices with any of its left lattices is torsionfree over the ground domain.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 11 , Issue 3 , December 1974 , pp. 365 - 371
Copyright
Copyright © Australian Mathematical Society 1974

References

[1]Bass, Hyman, “Injective dimension in Noetherian rings”, Trans. Amer. Math. Soc. 102 (1962), 18–29.CrossRefGoogle Scholar
[2]Дрозд, Ю. А., Кириченко, В. В., Ройтер, А. В. [Ju.A. Drozd, V.V. Kiričenko, A.V. Roi˘ter], “О наследстввнных и бассовых порядцах” [On hereditary and Bass orders], Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 1415–1436; English translation: Math. USSR Izv. 1 (1967), 1357–1376.Google Scholar
[3]Green, E. and Gustafson, W., “Pathological quasi-Frobenius algebras of finite type”, Comm. Algebra (to appear).Google Scholar
[4]Hattori, Akira, “On Prüfer rings”, J. Math. Soc. Japan 9 (1957), 381–385.Google Scholar
[5]Heller, A. and Reiner, I., “Representations of cyclic groups in rings of integers I”, Ann. of Math. (2) 76 (1962), 73–92.Google Scholar
[6]Яковлвв, A.B. [A. Jakovlev], “Гомологическая определенноть p–адических представлений колец со степенным базисом” [Homological determination of p–adic representations of rings with power basis], Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970), 1000–1014; English translation: Math. USSR Izv. 4 (1970), 1001–1016.Google Scholar
[7]Ройтер, A.B. [Roîter, A.V.], “Неограниченность размерностей неразложимых представлений алгебры, имеющей бесконечно много неразложимых представлений” [Unboundedness of the dimensions of indecomposable representations of algebras having infinitely many indecomposable representations], Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), 1275–1282; English translation: Math. USSR Izv. 2 (1968), 1223–1230.Google Scholar
[8]Rotman, Joseph J., Notes on homological algebra (Van Nostrand Reinhold Mathematical Studies, 27. Van Nostrand Reinhold Company, London, New York, Cincinnati, Toronto, Melbourne, 1970).Google Scholar
[9]Swan, Richard E., K-theory of finite groups and orders (notes by E. Graham Evans. Lecture Notes in Mathematics, 149. Springer-Verlag, Berlin, Heidelberg, New York, 1970).Google Scholar