On the Class Number of Representations of an Order

Abstract

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.