Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-08T00:22:59.275Z Has data issue: false hasContentIssue false
  • English
  • Français

II-Principal Hereditary Orders

Published online by Cambridge University Press:  22 January 2016

Susan Williamson
Susan Williamson*
Affiliation:
Regis College, Weston, Massachusetts
Rights & Permissions [Opens in a new window]

Extract

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 S denote the integral closure of a complete discrete rank one valuation ring R in a finite Galois extension of the quotient field of R, G the Galois group of the quotient field extension, and f an element of Z2(G,U(S)) where U(S) denotes the multiplicative group of units of S. A crossed product Δ(f, S, G) whose radical is generated as a left ideal by the prime element II of S is an hereditary order according to the Corollary to Thm. 2. 2 of [2], and we call such a crossed product a II-principal hereditary order. In previous papers the author has studied II-principal hereditary orders Δ(f, S, G) for tamely and wildly ramified extensions S of R (see [10] and [11]). The purpose of this paper is to study II-principal hereditary orders Δ(f, S, G) with no restriction on the extension S of R.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 32 , June 1968 , pp. 41 - 65
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1968

References

[1] Artin, E., Nesbitt, C. and Thrall, R.: Rings with Minimum Condition, Michigan, (1955).Google Scholar
[2] Auslander, M. and Goldman, O.: Maximal orders, Trans. Amer. Math. Soc. Vol. 97 (1960) pp. 1–24.Google Scholar
[3] Curtis, C. and Reiner, I.: Representation Theory of Finite Groups and Associative Algebras, Wiley, (1962).Google Scholar
[4] Hall, M.: The Theory of Groups, Macmillan, (1959).Google Scholar
[5] Harada, M.: Hereditary orders, Trans. Amer. Math. Soc. Vol. 107 (1963) pp. 273–290.CrossRefGoogle Scholar
[6] Harada, M.: Some criteria for hereditarity of crossed products, Osaka Math. J. Vol. 1 (1964) pp. 69–80.Google Scholar
[7] Serre, J.P.: Corps Locaux, Paris, Hermann, (1962).Google Scholar
[8] Waerden, B.L. van der: Modern Algebra, Vol. I, Ungar, (1953).Google Scholar
[9] Weiss, E.: Algebraic Number Theory, McGraw-Hill, (1963).Google Scholar
[10] Williamson, S.: Crossed products and hereditary orders, Nagoya Math. J. Vol. 23 (1963) pp. 103–120.CrossRefGoogle Scholar
[11] Williamson, S.: Crossed products and ramification, Nagoya Math. J. Vol. 28 (1966) pp. 85–111.Google Scholar