On generalized integral representations over Dedekind rings

The present paper develops the ideas presented in [1]. Let O be a Dedeking ring, and let Λ be a finitely generated algebra over O. An integral representation in the broad sense of the ring Λ over O is a homomorphism of Λ to the endomorphism ring of a finitely generated module over O. A representation in the restricted sense is a representation by matrices over O. Thus, the problem of describing the integral representations over O is subdivided into the following two problems: the description of representations in the broad sense and the selection of them of representations in the restricted sense. It is proved that any representation of A by matrices over the field k of fractions of the ring O is equivalent over k to an integral representation in the broad sense. This fact simplifies the problem of describing the representations in the broad sense. A representation is equivalent to a representation in the restricted sense if its degree over k and the order of the ideal class group of the ring O are relatively prime, or if it is the direct sum of h copies of one and the same representation over k, where h is the exponent of the ideal class group of O. ©1998 Plenum Publishing Corporation.