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On the Class Number of Representations of an Order

Published online by Cambridge University Press:  20 November 2018

Irving Reiner
Irving Reiner*
Affiliation:
University of Illinois
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We shall use the following notation throughout:

R = Dedekind ring (5).

u = multiplicative group of units in R.

h = class number of R.

K = quotient field of R.

p = prime ideal in R.

Rp = ring of p-adic integers in K.

We assume that h is finite, and that for each prime ideal p, the index (R:p) is finite.

Let A be a finite-dimensional separable algebra over K, with an identity element e (4, p. 115). Let G be an R-ordev in A, that is, G is a subring of A satisfying

  • (i) e ∈ G,

  • (ii) G contains a i∈-basis of A,

  • (iii) G is a finitely-generated i?-module.

By a G-module we shall mean a left G-module which is a finitely-generated torsion-free i∈-module, on which e acts as identity operator.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 11 , 1959 , pp. 660 - 672
Copyright
Copyright © Canadian Mathematical Society 1959

References

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