The rank of a completion of a dedekind domain |
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Authors: | Frank Okoh |
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Institution: | Department of Mathematics , Wayne State University , Detroit, Michigan, 48202 |
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Abstract: | Let D be a Dedekind domain with quotient field K. Let Cp be the completion of the localisationDp , of D at a nonzero prime idealp, of D. Let rp be the rank of Cp as a D-module, ierp , is the dimension of the K-vector space Kp , = K? DCp . The following results on rp are deduced from well-known theorems: if rp is finite for at least one prime ideal p, then D is a discrete valuation ring; and D = Cp if p = 1. If D is a discrete valuation ring, then rp = dimExt(K, D) + 1. A module M is extensionless if every extension of M by M splits. The D-module rC is an estensionless indecomposable module. If rC is infinite for every nonzero prime ideal, it is shown that an estensionless D-module of finite rank is a direct sum or certain rank one modulcs. |
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