Endomorphisms of modules over discrete valuation domains

Abstract

The primary objective of this dissertation is to study isomorphism theorems (or isomorphism problems) for modules over either complete or non-complete discrete valuation domains. The first chapter of the dissertation lays the foundation for all the remaining chapters. We first review the definition and some important facts about modules over discrete valuation domains. p-adic topologies and the notion of complete modules are then introduced, and indecomposable modules are examined. Then, we list some facts on homomorphism algebras of modules over discrete valuation domains that are necessary for our later studies. In the second chapter, we consider all possible isomorphism problems of modules over a complete discrete valuation domain. Kaplansky's and Wolfson's Theorems are presented, and we show that the isomorphism problem of modules over complete discrete valuation domains can be reduced to the isomorphism problem of reduced modules M with M/tM divisible. Finally, in the last chapter, we study isomorphism theorems for modules over a general discrete valuation domain. We present an isomorphism theorem for a special class of modules over a discrete valuation domain such that if M is in the class then any isomorphism of EndR(M) and EndR is induced by an isomorphism of M with M*